Distance on a great circle

Percentage Accurate: 62.0% → 98.6%

Time: 41.1s

Alternatives: 44

Speedup: 1.4×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 62.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.6% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.0%

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

   1. lift-sin.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift--.f64 N/A

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

   4. sub-flip N/A

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

   5. div-add N/A

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

   6. sin-sum N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-sin.f64 N/A

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

   9. mult-flip N/A

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

   10. metadata-eval N/A

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

   11. lower-*.f64 N/A

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

   12. lower-cos.f64 N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

   15. frac-2neg-rev N/A

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

   16. lower-/.f64 N/A

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

   17. metadata-eval N/A

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

   18. lower-*.f64 N/A

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

   19. lower-cos.f64 N/A

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

   20. mult-flip N/A

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

   21. metadata-eval N/A

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

   22. lower-*.f64 N/A

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

   23. lower-sin.f64 N/A

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

   24. metadata-eval N/A

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

   25. metadata-eval N/A

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

3. Applied rewrites 61.1%

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

   1. lift-sin.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift--.f64 N/A

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

   4. sub-flip N/A

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

   5. div-add N/A

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

   6. sin-sum N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-sin.f64 N/A

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

   9. mult-flip N/A

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

   10. metadata-eval N/A

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

   11. lower-*.f64 N/A

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

   12. lower-cos.f64 N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

   15. frac-2neg-rev N/A

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

   16. lower-/.f64 N/A

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

   17. metadata-eval N/A

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

   18. lower-*.f64 N/A

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

   19. lower-cos.f64 N/A

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

   20. mult-flip N/A

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

   21. metadata-eval N/A

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

   22. lower-*.f64 N/A

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

   23. lower-sin.f64 N/A

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

   24. metadata-eval N/A

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

   25. metadata-eval N/A

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

5. Applied rewrites 62.4%

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

   1. lift-sin.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift--.f64 N/A

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

   4. sub-flip N/A

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

   5. div-add N/A

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

   6. sin-sum N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-sin.f64 N/A

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

   9. mult-flip N/A

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

   10. metadata-eval N/A

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

   11. lower-*.f64 N/A

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

   12. lower-cos.f64 N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

   15. frac-2neg-rev N/A

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

   16. lower-/.f64 N/A

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

   17. metadata-eval N/A

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

   18. lower-*.f64 N/A

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

   19. lower-cos.f64 N/A

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

   20. mult-flip N/A

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

   21. metadata-eval N/A

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

   22. lower-*.f64 N/A

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

   23. lower-sin.f64 N/A

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

   24. metadata-eval N/A

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

   25. metadata-eval N/A

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

7. Applied rewrites 62.2%

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

   1. lift-sin.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift--.f64 N/A

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

   4. sub-flip N/A

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

   5. div-add N/A

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

   6. sin-sum N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-sin.f64 N/A

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

   9. mult-flip N/A

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

   10. metadata-eval N/A

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

   11. lower-*.f64 N/A

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

   12. lower-cos.f64 N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

   15. frac-2neg-rev N/A

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

   16. lower-/.f64 N/A

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

   17. metadata-eval N/A

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

   18. lower-*.f64 N/A

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

   19. lower-cos.f64 N/A

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

   20. mult-flip N/A

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

   21. metadata-eval N/A

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

   22. lower-*.f64 N/A

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

   23. lower-sin.f64 N/A

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

   24. metadata-eval N/A

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

   25. metadata-eval N/A

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

9. Applied rewrites 77.3%

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

    1. lift-sin.f64 N/A

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

    2. lift-/.f64 N/A

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

    3. lift--.f64 N/A

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

    4. div-sub N/A

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

    5. sin-diff N/A

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

    6. lower--.f64 N/A

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

    7. mult-flip-rev N/A

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

    8. metadata-eval N/A

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

    9. *-commutative N/A

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

    10. lift-*.f64 N/A

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

    11. lift-sin.f64 N/A

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

    12. lower-*.f64 N/A

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

    13. lower-cos.f64 N/A

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

    14. mult-flip N/A

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

    15. metadata-eval N/A

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

    16. lower-*.f64 N/A

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

    17. lower-*.f64 N/A

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

    18. lower-cos.f64 N/A

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

    19. mult-flip-rev N/A

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

    20. metadata-eval N/A

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

    21. *-commutative N/A

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

    22. lift-*.f64 N/A

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

    23. lower-sin.f64 N/A

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

    24. mult-flip N/A

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

    25. metadata-eval N/A

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

11. Applied rewrites 78.2%

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

    1. lift-sin.f64 N/A

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

    2. lift-/.f64 N/A

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

    3. lift--.f64 N/A

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

    4. div-sub N/A

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

    5. sin-diff N/A

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

    6. lower--.f64 N/A

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

    7. mult-flip-rev N/A

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

    8. metadata-eval N/A

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

    9. *-commutative N/A

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

    10. lift-*.f64 N/A

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

    11. lift-sin.f64 N/A

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

    12. lower-*.f64 N/A

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

    13. lower-cos.f64 N/A

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

    14. mult-flip N/A

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

    15. metadata-eval N/A

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

    16. lower-*.f64 N/A

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

    17. lower-*.f64 N/A

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

    18. lower-cos.f64 N/A

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

    19. mult-flip-rev N/A

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

    20. metadata-eval N/A

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

    21. *-commutative N/A

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

    22. lift-*.f64 N/A

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

    23. lower-sin.f64 N/A

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

    24. mult-flip N/A

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

    25. metadata-eval N/A

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

13. Applied rewrites 98.6%

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

14. Taylor expanded in lambda1 around inf

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

    1. lower-*.f64 N/A

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

    2. lower-cos.f64 N/A

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

    3. lower-*.f64 N/A

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

    4. lower-cos.f64 N/A

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

    5. lower-pow.f64 N/A

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

16. Applied rewrites 98.6%

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

17. Taylor expanded in lambda1 around inf

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

    1. lower-*.f64 N/A

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

    2. lower-cos.f64 N/A

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

    3. lower-*.f64 N/A

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

    4. lower-cos.f64 N/A

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

    5. lower-pow.f64 N/A

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

19. Applied rewrites 98.6%

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

    1. lift--.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. fp-cancel-sub-sign-inv N/A

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

    4. lift-*.f64 N/A

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

    5. lower-fma.f64 N/A

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

    6. lift-*.f64 N/A

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

    7. *-commutative N/A

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

    8. lower-*.f64 N/A

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

    9. lift-cos.f64 N/A

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

    10. cos-neg-rev N/A

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

    11. lower-cos.f64 N/A

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

    12. lift-*.f64 N/A

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

    13. *-commutative N/A

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

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

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

    15. metadata-eval N/A

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

    16. lower-*.f64 N/A

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

    17. lift-*.f64 N/A

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

    18. *-commutative N/A

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

    19. lift-*.f64 N/A

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

    20. lower-*.f64 N/A

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

21. Applied rewrites 98.6%

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

    1. lift--.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. fp-cancel-sub-sign-inv N/A

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

    4. lift-*.f64 N/A

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

    5. lower-fma.f64 N/A

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

    6. lift-*.f64 N/A

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

    7. *-commutative N/A

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

    8. lower-*.f64 N/A

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

    9. lift-cos.f64 N/A

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

    10. cos-neg-rev N/A

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

    11. lower-cos.f64 N/A

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

    12. lift-*.f64 N/A

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

    13. *-commutative N/A

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

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

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

    15. metadata-eval N/A

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

    16. lower-*.f64 N/A

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

    17. lift-*.f64 N/A

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

    18. *-commutative N/A

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

    19. lift-*.f64 N/A

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

    20. lower-*.f64 N/A

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

23. Applied rewrites 98.6%

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

Alternative 2: 98.6% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.0%

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

   1. lift-sin.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift--.f64 N/A

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

   4. sub-flip N/A

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

   5. div-add N/A

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

   6. sin-sum N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-sin.f64 N/A

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

   9. mult-flip N/A

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

   10. metadata-eval N/A

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

   11. lower-*.f64 N/A

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

   12. lower-cos.f64 N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

   15. frac-2neg-rev N/A

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

   16. lower-/.f64 N/A

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

   17. metadata-eval N/A

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

   18. lower-*.f64 N/A

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

   19. lower-cos.f64 N/A

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

   20. mult-flip N/A

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

   21. metadata-eval N/A

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

   22. lower-*.f64 N/A

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

   23. lower-sin.f64 N/A

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

   24. metadata-eval N/A

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

   25. metadata-eval N/A

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

3. Applied rewrites 61.1%

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

   1. lift-sin.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift--.f64 N/A

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

   4. sub-flip N/A

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

   5. div-add N/A

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

   6. sin-sum N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-sin.f64 N/A

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

   9. mult-flip N/A

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

   10. metadata-eval N/A

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

   11. lower-*.f64 N/A

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

   12. lower-cos.f64 N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

   15. frac-2neg-rev N/A

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

   16. lower-/.f64 N/A

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

   17. metadata-eval N/A

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

   18. lower-*.f64 N/A

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

   19. lower-cos.f64 N/A

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

   20. mult-flip N/A

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

   21. metadata-eval N/A

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

   22. lower-*.f64 N/A

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

   23. lower-sin.f64 N/A

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

   24. metadata-eval N/A

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

   25. metadata-eval N/A

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

5. Applied rewrites 62.4%

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

   1. lift-sin.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift--.f64 N/A

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

   4. sub-flip N/A

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

   5. div-add N/A

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

   6. sin-sum N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-sin.f64 N/A

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

   9. mult-flip N/A

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

   10. metadata-eval N/A

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

   11. lower-*.f64 N/A

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

   12. lower-cos.f64 N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

   15. frac-2neg-rev N/A

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

   16. lower-/.f64 N/A

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

   17. metadata-eval N/A

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

   18. lower-*.f64 N/A

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

   19. lower-cos.f64 N/A

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

   20. mult-flip N/A

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

   21. metadata-eval N/A

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

   22. lower-*.f64 N/A

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

   23. lower-sin.f64 N/A

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

   24. metadata-eval N/A

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

   25. metadata-eval N/A

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

7. Applied rewrites 62.2%

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

   1. lift-sin.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift--.f64 N/A

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

   4. sub-flip N/A

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

   5. div-add N/A

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

   6. sin-sum N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-sin.f64 N/A

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

   9. mult-flip N/A

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

   10. metadata-eval N/A

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

   11. lower-*.f64 N/A

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

   12. lower-cos.f64 N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

   15. frac-2neg-rev N/A

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

   16. lower-/.f64 N/A

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

   17. metadata-eval N/A

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

   18. lower-*.f64 N/A

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

   19. lower-cos.f64 N/A

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

   20. mult-flip N/A

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

   21. metadata-eval N/A

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

   22. lower-*.f64 N/A

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

   23. lower-sin.f64 N/A

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

   24. metadata-eval N/A

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

   25. metadata-eval N/A

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

9. Applied rewrites 77.3%

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

    1. lift-sin.f64 N/A

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

    2. lift-/.f64 N/A

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

    3. lift--.f64 N/A

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

    4. div-sub N/A

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

    5. sin-diff N/A

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

    6. lower--.f64 N/A

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

    7. mult-flip-rev N/A

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

    8. metadata-eval N/A

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

    9. *-commutative N/A

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

    10. lift-*.f64 N/A

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

    11. lift-sin.f64 N/A

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

    12. lower-*.f64 N/A

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

    13. lower-cos.f64 N/A

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

    14. mult-flip N/A

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

    15. metadata-eval N/A

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

    16. lower-*.f64 N/A

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

    17. lower-*.f64 N/A

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

    18. lower-cos.f64 N/A

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

    19. mult-flip-rev N/A

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

    20. metadata-eval N/A

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

    21. *-commutative N/A

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

    22. lift-*.f64 N/A

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

    23. lower-sin.f64 N/A

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

    24. mult-flip N/A

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

    25. metadata-eval N/A

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

11. Applied rewrites 78.2%

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

    1. lift-sin.f64 N/A

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

    2. lift-/.f64 N/A

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

    3. lift--.f64 N/A

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

    4. div-sub N/A

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

    5. sin-diff N/A

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

    6. lower--.f64 N/A

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

    7. mult-flip-rev N/A

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

    8. metadata-eval N/A

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

    9. *-commutative N/A

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

    10. lift-*.f64 N/A

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

    11. lift-sin.f64 N/A

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

    12. lower-*.f64 N/A

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

    13. lower-cos.f64 N/A

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

    14. mult-flip N/A

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

    15. metadata-eval N/A

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

    16. lower-*.f64 N/A

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

    17. lower-*.f64 N/A

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

    18. lower-cos.f64 N/A

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

    19. mult-flip-rev N/A

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

    20. metadata-eval N/A

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

    21. *-commutative N/A

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

    22. lift-*.f64 N/A

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

    23. lower-sin.f64 N/A

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

    24. mult-flip N/A

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

    25. metadata-eval N/A

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

13. Applied rewrites 98.6%

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

14. Taylor expanded in lambda1 around inf

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

    1. lower-*.f64 N/A

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

    2. lower-cos.f64 N/A

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

    3. lower-*.f64 N/A

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

    4. lower-cos.f64 N/A

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

    5. lower-pow.f64 N/A

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

16. Applied rewrites 98.6%

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

17. Taylor expanded in lambda1 around inf

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

    1. lower-*.f64 N/A

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

    2. lower-cos.f64 N/A

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

    3. lower-*.f64 N/A

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

    4. lower-cos.f64 N/A

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

    5. lower-pow.f64 N/A

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

19. Applied rewrites 98.6%

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

Alternative 3: 98.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(0.5 \cdot \phi_1\right)\\ t_1 := \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}\\ t_2 := \sin \left(0.5 \cdot \phi_1\right)\\ R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(t\_2 \cdot \cos \left(\phi_2 \cdot 0.5\right) - t\_0 \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \cos \phi_1 \cdot t\_1}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, t\_1, {\left(\cos \left(0.5 \cdot \phi_2\right) \cdot t\_2 - t\_0 \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\right)}}\right) \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (* 0.5 phi1)))
        (t_1
         (*
          (cos phi2)
          (pow
           (fma
            (cos (* -0.5 lambda2))
            (sin (* 0.5 lambda1))
            (* (cos (* 0.5 lambda1)) (sin (* -0.5 lambda2))))
           2.0)))
        (t_2 (sin (* 0.5 phi1))))
   (*
    R
    (*
     2.0
     (atan2
      (sqrt
       (+
        (pow (- (* t_2 (cos (* phi2 0.5))) (* t_0 (sin (* phi2 0.5)))) 2.0)
        (* (cos phi1) t_1)))
      (sqrt
       (-
        1.0
        (fma
         (cos phi1)
         t_1
         (pow
          (- (* (cos (* 0.5 phi2)) t_2) (* t_0 (sin (* 0.5 phi2))))
          2.0)))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((0.5 * phi1));
	double t_1 = cos(phi2) * pow(fma(cos((-0.5 * lambda2)), sin((0.5 * lambda1)), (cos((0.5 * lambda1)) * sin((-0.5 * lambda2)))), 2.0);
	double t_2 = sin((0.5 * phi1));
	return R * (2.0 * atan2(sqrt((pow(((t_2 * cos((phi2 * 0.5))) - (t_0 * sin((phi2 * 0.5)))), 2.0) + (cos(phi1) * t_1))), sqrt((1.0 - fma(cos(phi1), t_1, pow(((cos((0.5 * phi2)) * t_2) - (t_0 * sin((0.5 * phi2)))), 2.0))))));
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(0.5 * phi1))
	t_1 = Float64(cos(phi2) * (fma(cos(Float64(-0.5 * lambda2)), sin(Float64(0.5 * lambda1)), Float64(cos(Float64(0.5 * lambda1)) * sin(Float64(-0.5 * lambda2)))) ^ 2.0))
	t_2 = sin(Float64(0.5 * phi1))
	return Float64(R * Float64(2.0 * atan(sqrt(Float64((Float64(Float64(t_2 * cos(Float64(phi2 * 0.5))) - Float64(t_0 * sin(Float64(phi2 * 0.5)))) ^ 2.0) + Float64(cos(phi1) * t_1))), sqrt(Float64(1.0 - fma(cos(phi1), t_1, (Float64(Float64(cos(Float64(0.5 * phi2)) * t_2) - Float64(t_0 * sin(Float64(0.5 * phi2)))) ^ 2.0)))))))
end

Wolfram

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

Derivation

1. Initial program 62.0%

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

   1. lift-sin.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift--.f64 N/A

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

   4. sub-flip N/A

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

   5. div-add N/A

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

   6. sin-sum N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-sin.f64 N/A

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

   9. mult-flip N/A

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

   10. metadata-eval N/A

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

   11. lower-*.f64 N/A

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

   12. lower-cos.f64 N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

   15. frac-2neg-rev N/A

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

   16. lower-/.f64 N/A

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

   17. metadata-eval N/A

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

   18. lower-*.f64 N/A

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

   19. lower-cos.f64 N/A

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

   20. mult-flip N/A

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

   21. metadata-eval N/A

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

   22. lower-*.f64 N/A

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

   23. lower-sin.f64 N/A

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

   24. metadata-eval N/A

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

   25. metadata-eval N/A

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

3. Applied rewrites 61.1%

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

   1. lift-sin.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift--.f64 N/A

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

   4. sub-flip N/A

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

   5. div-add N/A

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

   6. sin-sum N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-sin.f64 N/A

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

   9. mult-flip N/A

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

   10. metadata-eval N/A

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

   11. lower-*.f64 N/A

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

   12. lower-cos.f64 N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

   15. frac-2neg-rev N/A

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

   16. lower-/.f64 N/A

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

   17. metadata-eval N/A

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

   18. lower-*.f64 N/A

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

   19. lower-cos.f64 N/A

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

   20. mult-flip N/A

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

   21. metadata-eval N/A

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

   22. lower-*.f64 N/A

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

   23. lower-sin.f64 N/A

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

   24. metadata-eval N/A

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

   25. metadata-eval N/A

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

5. Applied rewrites 62.4%

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

   1. lift-sin.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift--.f64 N/A

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

   4. sub-flip N/A

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

   5. div-add N/A

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

   6. sin-sum N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-sin.f64 N/A

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

   9. mult-flip N/A

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

   10. metadata-eval N/A

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

   11. lower-*.f64 N/A

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

   12. lower-cos.f64 N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

   15. frac-2neg-rev N/A

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

   16. lower-/.f64 N/A

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

   17. metadata-eval N/A

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

   18. lower-*.f64 N/A

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

   19. lower-cos.f64 N/A

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

   20. mult-flip N/A

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

   21. metadata-eval N/A

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

   22. lower-*.f64 N/A

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

   23. lower-sin.f64 N/A

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

   24. metadata-eval N/A

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

   25. metadata-eval N/A

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

7. Applied rewrites 62.2%

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

   1. lift-sin.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift--.f64 N/A

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

   4. sub-flip N/A

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

   5. div-add N/A

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

   6. sin-sum N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-sin.f64 N/A

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

   9. mult-flip N/A

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

   10. metadata-eval N/A

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

   11. lower-*.f64 N/A

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

   12. lower-cos.f64 N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

   15. frac-2neg-rev N/A

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

   16. lower-/.f64 N/A

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

   17. metadata-eval N/A

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

   18. lower-*.f64 N/A

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

   19. lower-cos.f64 N/A

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

   20. mult-flip N/A

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

   21. metadata-eval N/A

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

   22. lower-*.f64 N/A

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

   23. lower-sin.f64 N/A

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

   24. metadata-eval N/A

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

   25. metadata-eval N/A

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

9. Applied rewrites 77.3%

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

    1. lift-sin.f64 N/A

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

    2. lift-/.f64 N/A

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

    3. lift--.f64 N/A

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

    4. div-sub N/A

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

    5. sin-diff N/A

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

    6. lower--.f64 N/A

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

    7. mult-flip-rev N/A

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

    8. metadata-eval N/A

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

    9. *-commutative N/A

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

    10. lift-*.f64 N/A

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

    11. lift-sin.f64 N/A

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

    12. lower-*.f64 N/A

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

    13. lower-cos.f64 N/A

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

    14. mult-flip N/A

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

    15. metadata-eval N/A

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

    16. lower-*.f64 N/A

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

    17. lower-*.f64 N/A

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

    18. lower-cos.f64 N/A

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

    19. mult-flip-rev N/A

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

    20. metadata-eval N/A

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

    21. *-commutative N/A

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

    22. lift-*.f64 N/A

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

    23. lower-sin.f64 N/A

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

    24. mult-flip N/A

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

    25. metadata-eval N/A

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

11. Applied rewrites 78.2%

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

    1. lift-sin.f64 N/A

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

    2. lift-/.f64 N/A

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

    3. lift--.f64 N/A

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

    4. div-sub N/A

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

    5. sin-diff N/A

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

    6. lower--.f64 N/A

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

    7. mult-flip-rev N/A

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

    8. metadata-eval N/A

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

    9. *-commutative N/A

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

    10. lift-*.f64 N/A

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

    11. lift-sin.f64 N/A

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

    12. lower-*.f64 N/A

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

    13. lower-cos.f64 N/A

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

    14. mult-flip N/A

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

    15. metadata-eval N/A

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

    16. lower-*.f64 N/A

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

    17. lower-*.f64 N/A

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

    18. lower-cos.f64 N/A

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

    19. mult-flip-rev N/A

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

    20. metadata-eval N/A

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

    21. *-commutative N/A

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

    22. lift-*.f64 N/A

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

    23. lower-sin.f64 N/A

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

    24. mult-flip N/A

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

    25. metadata-eval N/A

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

13. Applied rewrites 98.6%

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

14. Taylor expanded in lambda1 around inf

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

    1. lower-*.f64 N/A

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

    2. lower-cos.f64 N/A

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

    3. lower-*.f64 N/A

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

    4. lower-cos.f64 N/A

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

    5. lower-pow.f64 N/A

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

16. Applied rewrites 98.6%

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

17. Taylor expanded in lambda1 around inf

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

    1. lower-*.f64 N/A

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

    2. lower-cos.f64 N/A

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

    3. lower-*.f64 N/A

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

    4. lower-cos.f64 N/A

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

    5. lower-pow.f64 N/A

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

19. Applied rewrites 98.6%

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

20. Taylor expanded in lambda1 around inf

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

22. Applied rewrites 98.6%

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

Alternative 4: 98.6% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.0%

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

   1. lift-sin.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift--.f64 N/A

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

   4. sub-flip N/A

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

   5. div-add N/A

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

   6. sin-sum N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-sin.f64 N/A

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

   9. mult-flip N/A

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

   10. metadata-eval N/A

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

   11. lower-*.f64 N/A

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

   12. lower-cos.f64 N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

   15. frac-2neg-rev N/A

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

   16. lower-/.f64 N/A

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

   17. metadata-eval N/A

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

   18. lower-*.f64 N/A

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

   19. lower-cos.f64 N/A

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

   20. mult-flip N/A

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

   21. metadata-eval N/A

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

   22. lower-*.f64 N/A

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

   23. lower-sin.f64 N/A

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

   24. metadata-eval N/A

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

   25. metadata-eval N/A

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

3. Applied rewrites 61.1%

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

   1. lift-sin.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift--.f64 N/A

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

   4. sub-flip N/A

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

   5. div-add N/A

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

   6. sin-sum N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-sin.f64 N/A

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

   9. mult-flip N/A

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

   10. metadata-eval N/A

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

   11. lower-*.f64 N/A

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

   12. lower-cos.f64 N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

   15. frac-2neg-rev N/A

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

   16. lower-/.f64 N/A

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

   17. metadata-eval N/A

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

   18. lower-*.f64 N/A

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

   19. lower-cos.f64 N/A

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

   20. mult-flip N/A

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

   21. metadata-eval N/A

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

   22. lower-*.f64 N/A

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

   23. lower-sin.f64 N/A

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

   24. metadata-eval N/A

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

   25. metadata-eval N/A

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

5. Applied rewrites 62.4%

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

   1. lift-sin.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift--.f64 N/A

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

   4. sub-flip N/A

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

   5. div-add N/A

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

   6. sin-sum N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-sin.f64 N/A

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

   9. mult-flip N/A

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

   10. metadata-eval N/A

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

   11. lower-*.f64 N/A

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

   12. lower-cos.f64 N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

   15. frac-2neg-rev N/A

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

   16. lower-/.f64 N/A

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

   17. metadata-eval N/A

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

   18. lower-*.f64 N/A

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

   19. lower-cos.f64 N/A

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

   20. mult-flip N/A

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

   21. metadata-eval N/A

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

   22. lower-*.f64 N/A

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

   23. lower-sin.f64 N/A

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

   24. metadata-eval N/A

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

   25. metadata-eval N/A

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

7. Applied rewrites 62.2%

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

   1. lift-sin.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift--.f64 N/A

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

   4. sub-flip N/A

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

   5. div-add N/A

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

   6. sin-sum N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-sin.f64 N/A

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

   9. mult-flip N/A

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

   10. metadata-eval N/A

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

   11. lower-*.f64 N/A

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

   12. lower-cos.f64 N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

   15. frac-2neg-rev N/A

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

   16. lower-/.f64 N/A

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

   17. metadata-eval N/A

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

   18. lower-*.f64 N/A

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

   19. lower-cos.f64 N/A

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

   20. mult-flip N/A

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

   21. metadata-eval N/A

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

   22. lower-*.f64 N/A

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

   23. lower-sin.f64 N/A

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

   24. metadata-eval N/A

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

   25. metadata-eval N/A

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

9. Applied rewrites 77.3%

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

    1. lift-sin.f64 N/A

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

    2. lift-/.f64 N/A

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

    3. lift--.f64 N/A

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

    4. div-sub N/A

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

    5. sin-diff N/A

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

    6. lower--.f64 N/A

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

    7. mult-flip-rev N/A

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

    8. metadata-eval N/A

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

    9. *-commutative N/A

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

    10. lift-*.f64 N/A

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

    11. lift-sin.f64 N/A

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

    12. lower-*.f64 N/A

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

    13. lower-cos.f64 N/A

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

    14. mult-flip N/A

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

    15. metadata-eval N/A

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

    16. lower-*.f64 N/A

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

    17. lower-*.f64 N/A

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

    18. lower-cos.f64 N/A

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

    19. mult-flip-rev N/A

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

    20. metadata-eval N/A

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

    21. *-commutative N/A

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

    22. lift-*.f64 N/A

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

    23. lower-sin.f64 N/A

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

    24. mult-flip N/A

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

    25. metadata-eval N/A

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

11. Applied rewrites 78.2%

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

    1. lift-sin.f64 N/A

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

    2. lift-/.f64 N/A

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

    3. lift--.f64 N/A

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

    4. div-sub N/A

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

    5. sin-diff N/A

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

    6. lower--.f64 N/A

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

    7. mult-flip-rev N/A

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

    8. metadata-eval N/A

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

    9. *-commutative N/A

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

    10. lift-*.f64 N/A

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

    11. lift-sin.f64 N/A

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

    12. lower-*.f64 N/A

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

    13. lower-cos.f64 N/A

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

    14. mult-flip N/A

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

    15. metadata-eval N/A

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

    16. lower-*.f64 N/A

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

    17. lower-*.f64 N/A

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

    18. lower-cos.f64 N/A

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

    19. mult-flip-rev N/A

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

    20. metadata-eval N/A

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

    21. *-commutative N/A

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

    22. lift-*.f64 N/A

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

    23. lower-sin.f64 N/A

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

    24. mult-flip N/A

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

    25. metadata-eval N/A

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

13. Applied rewrites 98.6%

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

14. Taylor expanded in lambda1 around inf

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

    1. lower-fma.f64 N/A

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

16. Applied rewrites 98.6%

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

17. Taylor expanded in lambda1 around inf

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

    1. lower-fma.f64 N/A

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

19. Applied rewrites 98.6%

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

Alternative 5: 87.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := \sin \left(0.5 \cdot \lambda_1\right)\\ t_1 := \sin \left(-0.5 \cdot \lambda_2\right)\\ t_2 := \sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right)\\ t_3 := \cos \left(-0.5 \cdot \lambda_2\right)\\ t_4 := {\left(t\_2 - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(t\_3, t\_0, \left(1 + -0.125 \cdot {\lambda_1}^{2}\right) \cdot t\_1\right)\right)}^{2}\right)\\ t_5 := {\left(t\_2 - \sin \left(0.5 \cdot \phi_2\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(t\_3, t\_0, \cos \left(0.5 \cdot \lambda_1\right) \cdot t\_1\right)\right)}^{2}\right)\\ t_6 := R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_5}}{\sqrt{1 - t\_5}}\right)\\ \mathbf{if}\;\lambda_1 \leq -2700000:\\ \;\;\;\;t\_6\\ \mathbf{elif}\;\lambda_1 \leq 190:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_4}}{\sqrt{1 - t\_4}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_6\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (* 0.5 lambda1)))
        (t_1 (sin (* -0.5 lambda2)))
        (t_2 (* (sin (* 0.5 phi1)) (cos (* phi2 0.5))))
        (t_3 (cos (* -0.5 lambda2)))
        (t_4
         (+
          (pow (- t_2 (* (cos (* 0.5 phi1)) (sin (* phi2 0.5)))) 2.0)
          (*
           (cos phi1)
           (*
            (cos phi2)
            (pow
             (fma t_3 t_0 (* (+ 1.0 (* -0.125 (pow lambda1 2.0))) t_1))
             2.0)))))
        (t_5
         (+
          (pow (- t_2 (sin (* 0.5 phi2))) 2.0)
          (*
           (cos phi1)
           (*
            (cos phi2)
            (pow (fma t_3 t_0 (* (cos (* 0.5 lambda1)) t_1)) 2.0)))))
        (t_6 (* R (* 2.0 (atan2 (sqrt t_5) (sqrt (- 1.0 t_5)))))))
   (if (<= lambda1 -2700000.0)
     t_6
     (if (<= lambda1 190.0)
       (* R (* 2.0 (atan2 (sqrt t_4) (sqrt (- 1.0 t_4)))))
       t_6))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin((0.5 * lambda1));
	double t_1 = sin((-0.5 * lambda2));
	double t_2 = sin((0.5 * phi1)) * cos((phi2 * 0.5));
	double t_3 = cos((-0.5 * lambda2));
	double t_4 = pow((t_2 - (cos((0.5 * phi1)) * sin((phi2 * 0.5)))), 2.0) + (cos(phi1) * (cos(phi2) * pow(fma(t_3, t_0, ((1.0 + (-0.125 * pow(lambda1, 2.0))) * t_1)), 2.0)));
	double t_5 = pow((t_2 - sin((0.5 * phi2))), 2.0) + (cos(phi1) * (cos(phi2) * pow(fma(t_3, t_0, (cos((0.5 * lambda1)) * t_1)), 2.0)));
	double t_6 = R * (2.0 * atan2(sqrt(t_5), sqrt((1.0 - t_5))));
	double tmp;
	if (lambda1 <= -2700000.0) {
		tmp = t_6;
	} else if (lambda1 <= 190.0) {
		tmp = R * (2.0 * atan2(sqrt(t_4), sqrt((1.0 - t_4))));
	} else {
		tmp = t_6;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(0.5 * lambda1))
	t_1 = sin(Float64(-0.5 * lambda2))
	t_2 = Float64(sin(Float64(0.5 * phi1)) * cos(Float64(phi2 * 0.5)))
	t_3 = cos(Float64(-0.5 * lambda2))
	t_4 = Float64((Float64(t_2 - Float64(cos(Float64(0.5 * phi1)) * sin(Float64(phi2 * 0.5)))) ^ 2.0) + Float64(cos(phi1) * Float64(cos(phi2) * (fma(t_3, t_0, Float64(Float64(1.0 + Float64(-0.125 * (lambda1 ^ 2.0))) * t_1)) ^ 2.0))))
	t_5 = Float64((Float64(t_2 - sin(Float64(0.5 * phi2))) ^ 2.0) + Float64(cos(phi1) * Float64(cos(phi2) * (fma(t_3, t_0, Float64(cos(Float64(0.5 * lambda1)) * t_1)) ^ 2.0))))
	t_6 = Float64(R * Float64(2.0 * atan(sqrt(t_5), sqrt(Float64(1.0 - t_5)))))
	tmp = 0.0
	if (lambda1 <= -2700000.0)
		tmp = t_6;
	elseif (lambda1 <= 190.0)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_4), sqrt(Float64(1.0 - t_4)))));
	else
		tmp = t_6;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(-0.5 * lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Cos[N[(-0.5 * lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(N[Power[N[(t$95$2 - N[(N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Power[N[(t$95$3 * t$95$0 + N[(N[(1.0 + N[(-0.125 * N[Power[lambda1, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[Power[N[(t$95$2 - N[Sin[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Power[N[(t$95$3 * t$95$0 + N[(N[Cos[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$5], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[lambda1, -2700000.0], t$95$6, If[LessEqual[lambda1, 190.0], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$4], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$4), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$6]]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if lambda1 < -2.7e6 or 190 < lambda1

   1. Initial program 62.0%

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

      1. lift-sin.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. sub-flip N/A

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

      5. div-add N/A

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

      6. sin-sum N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. mult-flip N/A

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

      10. metadata-eval N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. frac-2neg-rev N/A

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

      16. lower-/.f64 N/A

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

      17. metadata-eval N/A

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

      18. lower-*.f64 N/A

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

      19. lower-cos.f64 N/A

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

      20. mult-flip N/A

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

      21. metadata-eval N/A

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

      22. lower-*.f64 N/A

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

      23. lower-sin.f64 N/A

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

      24. metadata-eval N/A

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

      25. metadata-eval N/A

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

   3. Applied rewrites 61.1%

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

      1. lift-sin.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. sub-flip N/A

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

      5. div-add N/A

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

      6. sin-sum N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. mult-flip N/A

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

      10. metadata-eval N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. frac-2neg-rev N/A

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

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 62.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 62.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}}\right) \]

   9. Applied rewrites 77.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right)}}\right) \]
   10. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       7. mult-flip-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       8. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       9. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       10. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       11. lift-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       12. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       13. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       14. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       15. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       17. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       18. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       19. mult-flip-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       20. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       21. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       22. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       23. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       24. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       25. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   11. Applied rewrites 78.2%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   12. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       7. mult-flip-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       8. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       9. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       10. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       11. lift-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       12. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       13. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       14. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       15. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       17. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       18. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       19. mult-flip-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       20. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       21. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       22. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       23. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       24. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       25. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   13. Applied rewrites 98.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   14. Taylor expanded in lambda1 around inf

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   15. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \color{blue}{\left(\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\color{blue}{\cos \phi_2} \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \color{blue}{{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\color{blue}{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{\color{blue}{2}}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   16. Applied rewrites 98.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   17. Taylor expanded in lambda1 around inf

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}\right)}\right)}}\right) \]
   18. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \color{blue}{\left(\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}\right)}\right)}}\right) \]

       2. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\color{blue}{\cos \phi_2} \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}\right)\right)}}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \color{blue}{{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}\right)\right)}}\right) \]

       4. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\color{blue}{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}}^{2}\right)\right)}}\right) \]

       5. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{\color{blue}{2}}\right)\right)}}\right) \]

   19. Applied rewrites 98.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}\right)}\right)}}\right) \]

   20. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\sin \left(\frac{1}{2} \cdot \phi_2\right)}\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]
   21. Step-by-step derivation

       1. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       2. lower-*.f64 79.0

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \sin \left(0.5 \cdot \phi_2\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

   22. Applied rewrites 79.0%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \color{blue}{\sin \left(0.5 \cdot \phi_2\right)}\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

   23. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\sin \left(\frac{1}{2} \cdot \phi_2\right)}\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]
   24. Step-by-step derivation

       1. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       2. lower-*.f64 76.9

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \sin \left(0.5 \cdot \phi_2\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \sin \left(0.5 \cdot \phi_2\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

   25. Applied rewrites 76.9%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \sin \left(0.5 \cdot \phi_2\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \color{blue}{\sin \left(0.5 \cdot \phi_2\right)}\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

   if -2.7e6 < lambda1 < 190

   1. Initial program 62.0%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 61.1%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 62.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 62.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}}\right) \]

   9. Applied rewrites 77.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right)}}\right) \]
   10. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       7. mult-flip-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       8. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       9. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       10. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       11. lift-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       12. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       13. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       14. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       15. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       17. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       18. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       19. mult-flip-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       20. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       21. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       22. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       23. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       24. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       25. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   11. Applied rewrites 78.2%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   12. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       7. mult-flip-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       8. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       9. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       10. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       11. lift-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       12. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       13. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       14. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       15. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       17. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       18. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       19. mult-flip-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       20. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       21. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       22. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       23. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       24. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       25. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   13. Applied rewrites 98.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   14. Taylor expanded in lambda1 around inf

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   15. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \color{blue}{\left(\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\color{blue}{\cos \phi_2} \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \color{blue}{{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\color{blue}{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{\color{blue}{2}}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   16. Applied rewrites 98.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   17. Taylor expanded in lambda1 around inf

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}\right)}\right)}}\right) \]
   18. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \color{blue}{\left(\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}\right)}\right)}}\right) \]

       2. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\color{blue}{\cos \phi_2} \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}\right)\right)}}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \color{blue}{{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}\right)\right)}}\right) \]

       4. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\color{blue}{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}}^{2}\right)\right)}}\right) \]

       5. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{\color{blue}{2}}\right)\right)}}\right) \]

   19. Applied rewrites 98.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}\right)}\right)}}\right) \]

   20. Taylor expanded in lambda1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \left(1 + \frac{-1}{8} \cdot {\lambda_1}^{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]
   21. Step-by-step derivation

       1. lower-+.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \left(1 + \frac{-1}{8} \cdot {\lambda_1}^{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       2. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \left(1 + \frac{-1}{8} \cdot {\lambda_1}^{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       3. lower-pow.f64 59.8

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \left(1 + -0.125 \cdot {\lambda_1}^{2}\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

   22. Applied rewrites 59.8%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \left(1 + -0.125 \cdot {\lambda_1}^{2}\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

   23. Taylor expanded in lambda1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \left(1 + \frac{-1}{8} \cdot {\lambda_1}^{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \left(1 + \frac{-1}{8} \cdot {\lambda_1}^{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]
   24. Step-by-step derivation

       1. lower-+.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \left(1 + \frac{-1}{8} \cdot {\lambda_1}^{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \left(1 + \frac{-1}{8} \cdot {\lambda_1}^{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       2. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \left(1 + \frac{-1}{8} \cdot {\lambda_1}^{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \left(1 + \frac{-1}{8} \cdot {\lambda_1}^{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       3. lower-pow.f64 54.5

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \left(1 + -0.125 \cdot {\lambda_1}^{2}\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \left(1 + -0.125 \cdot {\lambda_1}^{2}\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

   25. Applied rewrites 54.5%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \left(1 + -0.125 \cdot {\lambda_1}^{2}\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \left(1 + -0.125 \cdot {\lambda_1}^{2}\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 87.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_0 := \sin \left(0.5 \cdot \lambda_1\right)\\ t_1 := \sin \left(-0.5 \cdot \lambda_2\right)\\ t_2 := \sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right)\\ t_3 := \cos \left(-0.5 \cdot \lambda_2\right)\\ t_4 := {\left(t\_2 - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(t\_3, t\_0, 1 \cdot t\_1\right)\right)}^{2}\right)\\ t_5 := {\left(t\_2 - \sin \left(0.5 \cdot \phi_2\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(t\_3, t\_0, \cos \left(0.5 \cdot \lambda_1\right) \cdot t\_1\right)\right)}^{2}\right)\\ t_6 := R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_5}}{\sqrt{1 - t\_5}}\right)\\ \mathbf{if}\;\lambda_1 \leq -7.2 \cdot 10^{+31}:\\ \;\;\;\;t\_6\\ \mathbf{elif}\;\lambda_1 \leq 660:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_4}}{\sqrt{1 - t\_4}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_6\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (* 0.5 lambda1)))
        (t_1 (sin (* -0.5 lambda2)))
        (t_2 (* (sin (* 0.5 phi1)) (cos (* phi2 0.5))))
        (t_3 (cos (* -0.5 lambda2)))
        (t_4
         (+
          (pow (- t_2 (* (cos (* 0.5 phi1)) (sin (* phi2 0.5)))) 2.0)
          (* (cos phi1) (* (cos phi2) (pow (fma t_3 t_0 (* 1.0 t_1)) 2.0)))))
        (t_5
         (+
          (pow (- t_2 (sin (* 0.5 phi2))) 2.0)
          (*
           (cos phi1)
           (*
            (cos phi2)
            (pow (fma t_3 t_0 (* (cos (* 0.5 lambda1)) t_1)) 2.0)))))
        (t_6 (* R (* 2.0 (atan2 (sqrt t_5) (sqrt (- 1.0 t_5)))))))
   (if (<= lambda1 -7.2e+31)
     t_6
     (if (<= lambda1 660.0)
       (* R (* 2.0 (atan2 (sqrt t_4) (sqrt (- 1.0 t_4)))))
       t_6))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin((0.5 * lambda1));
	double t_1 = sin((-0.5 * lambda2));
	double t_2 = sin((0.5 * phi1)) * cos((phi2 * 0.5));
	double t_3 = cos((-0.5 * lambda2));
	double t_4 = pow((t_2 - (cos((0.5 * phi1)) * sin((phi2 * 0.5)))), 2.0) + (cos(phi1) * (cos(phi2) * pow(fma(t_3, t_0, (1.0 * t_1)), 2.0)));
	double t_5 = pow((t_2 - sin((0.5 * phi2))), 2.0) + (cos(phi1) * (cos(phi2) * pow(fma(t_3, t_0, (cos((0.5 * lambda1)) * t_1)), 2.0)));
	double t_6 = R * (2.0 * atan2(sqrt(t_5), sqrt((1.0 - t_5))));
	double tmp;
	if (lambda1 <= -7.2e+31) {
		tmp = t_6;
	} else if (lambda1 <= 660.0) {
		tmp = R * (2.0 * atan2(sqrt(t_4), sqrt((1.0 - t_4))));
	} else {
		tmp = t_6;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(0.5 * lambda1))
	t_1 = sin(Float64(-0.5 * lambda2))
	t_2 = Float64(sin(Float64(0.5 * phi1)) * cos(Float64(phi2 * 0.5)))
	t_3 = cos(Float64(-0.5 * lambda2))
	t_4 = Float64((Float64(t_2 - Float64(cos(Float64(0.5 * phi1)) * sin(Float64(phi2 * 0.5)))) ^ 2.0) + Float64(cos(phi1) * Float64(cos(phi2) * (fma(t_3, t_0, Float64(1.0 * t_1)) ^ 2.0))))
	t_5 = Float64((Float64(t_2 - sin(Float64(0.5 * phi2))) ^ 2.0) + Float64(cos(phi1) * Float64(cos(phi2) * (fma(t_3, t_0, Float64(cos(Float64(0.5 * lambda1)) * t_1)) ^ 2.0))))
	t_6 = Float64(R * Float64(2.0 * atan(sqrt(t_5), sqrt(Float64(1.0 - t_5)))))
	tmp = 0.0
	if (lambda1 <= -7.2e+31)
		tmp = t_6;
	elseif (lambda1 <= 660.0)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_4), sqrt(Float64(1.0 - t_4)))));
	else
		tmp = t_6;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(-0.5 * lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Cos[N[(-0.5 * lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(N[Power[N[(t$95$2 - N[(N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Power[N[(t$95$3 * t$95$0 + N[(1.0 * t$95$1), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[Power[N[(t$95$2 - N[Sin[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Power[N[(t$95$3 * t$95$0 + N[(N[Cos[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$5], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[lambda1, -7.2e+31], t$95$6, If[LessEqual[lambda1, 660.0], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$4], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$4), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$6]]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if lambda1 < -7.19999999999999992e31 or 660 < lambda1

   1. Initial program 62.0%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 61.1%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 62.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 62.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}}\right) \]

   9. Applied rewrites 77.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right)}}\right) \]
   10. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       7. mult-flip-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       8. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       9. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       10. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       11. lift-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       12. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       13. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       14. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       15. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       17. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       18. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       19. mult-flip-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       20. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       21. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       22. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       23. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       24. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       25. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   11. Applied rewrites 78.2%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   12. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       7. mult-flip-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       8. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       9. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       10. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       11. lift-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       12. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       13. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       14. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       15. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       17. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       18. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       19. mult-flip-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       20. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       21. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       22. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       23. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       24. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       25. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   13. Applied rewrites 98.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   14. Taylor expanded in lambda1 around inf

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   15. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \color{blue}{\left(\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\color{blue}{\cos \phi_2} \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \color{blue}{{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\color{blue}{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{\color{blue}{2}}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   16. Applied rewrites 98.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   17. Taylor expanded in lambda1 around inf

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}\right)}\right)}}\right) \]
   18. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \color{blue}{\left(\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}\right)}\right)}}\right) \]

       2. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\color{blue}{\cos \phi_2} \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}\right)\right)}}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \color{blue}{{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}\right)\right)}}\right) \]

       4. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\color{blue}{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}}^{2}\right)\right)}}\right) \]

       5. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{\color{blue}{2}}\right)\right)}}\right) \]

   19. Applied rewrites 98.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}\right)}\right)}}\right) \]

   20. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\sin \left(\frac{1}{2} \cdot \phi_2\right)}\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]
   21. Step-by-step derivation

       1. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       2. lower-*.f64 79.0

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \sin \left(0.5 \cdot \phi_2\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

   22. Applied rewrites 79.0%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \color{blue}{\sin \left(0.5 \cdot \phi_2\right)}\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

   23. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\sin \left(\frac{1}{2} \cdot \phi_2\right)}\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]
   24. Step-by-step derivation

       1. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       2. lower-*.f64 76.9

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \sin \left(0.5 \cdot \phi_2\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \sin \left(0.5 \cdot \phi_2\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

   25. Applied rewrites 76.9%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \sin \left(0.5 \cdot \phi_2\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \color{blue}{\sin \left(0.5 \cdot \phi_2\right)}\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

   if -7.19999999999999992e31 < lambda1 < 660

   1. Initial program 62.0%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 61.1%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 62.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 62.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}}\right) \]

   9. Applied rewrites 77.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right)}}\right) \]
   10. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       7. mult-flip-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       8. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       9. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       10. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       11. lift-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       12. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       13. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       14. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       15. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       17. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       18. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       19. mult-flip-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       20. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       21. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       22. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       23. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       24. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       25. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   11. Applied rewrites 78.2%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   12. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       7. mult-flip-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       8. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       9. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       10. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       11. lift-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       12. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       13. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       14. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       15. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       17. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       18. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       19. mult-flip-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       20. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       21. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       22. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       23. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       24. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       25. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   13. Applied rewrites 98.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   14. Taylor expanded in lambda1 around inf

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   15. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \color{blue}{\left(\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\color{blue}{\cos \phi_2} \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \color{blue}{{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\color{blue}{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{\color{blue}{2}}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   16. Applied rewrites 98.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   17. Taylor expanded in lambda1 around inf

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}\right)}\right)}}\right) \]
   18. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \color{blue}{\left(\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}\right)}\right)}}\right) \]

       2. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\color{blue}{\cos \phi_2} \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}\right)\right)}}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \color{blue}{{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}\right)\right)}}\right) \]

       4. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\color{blue}{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}}^{2}\right)\right)}}\right) \]

       5. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{\color{blue}{2}}\right)\right)}}\right) \]

   19. Applied rewrites 98.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}\right)}\right)}}\right) \]

   20. Taylor expanded in lambda1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), 1 \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]
   21. Step-by-step derivation

   22. Applied rewrites 78.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), 1 \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

   23. Taylor expanded in lambda1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), 1 \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), 1 \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]
   24. Step-by-step derivation

   25. Applied rewrites 77.2%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), 1 \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), 1 \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 87.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := {\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_0\right) \cdot t\_0\\ t_2 := {\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \sin \left(0.5 \cdot \phi_2\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}\right)\\ t_3 := R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_2}}{\sqrt{1 - t\_2}}\right)\\ \mathbf{if}\;\lambda_1 \leq -1.35 \cdot 10^{+34}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;\lambda_1 \leq 130000000:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_1}}{\sqrt{1 - t\_1}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_1
         (+
          (pow
           (fma
            (sin (* phi1 0.5))
            (cos (/ phi2 -2.0))
            (* (cos (* phi1 0.5)) (sin (/ phi2 -2.0))))
           2.0)
          (* (* (* (cos phi1) (cos phi2)) t_0) t_0)))
        (t_2
         (+
          (pow
           (- (* (sin (* 0.5 phi1)) (cos (* phi2 0.5))) (sin (* 0.5 phi2)))
           2.0)
          (*
           (cos phi1)
           (*
            (cos phi2)
            (pow
             (fma
              (cos (* -0.5 lambda2))
              (sin (* 0.5 lambda1))
              (* (cos (* 0.5 lambda1)) (sin (* -0.5 lambda2))))
             2.0)))))
        (t_3 (* R (* 2.0 (atan2 (sqrt t_2) (sqrt (- 1.0 t_2)))))))
   (if (<= lambda1 -1.35e+34)
     t_3
     (if (<= lambda1 130000000.0)
       (* R (* 2.0 (atan2 (sqrt t_1) (sqrt (- 1.0 t_1)))))
       t_3))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(((lambda1 - lambda2) / 2.0));
	double t_1 = pow(fma(sin((phi1 * 0.5)), cos((phi2 / -2.0)), (cos((phi1 * 0.5)) * sin((phi2 / -2.0)))), 2.0) + (((cos(phi1) * cos(phi2)) * t_0) * t_0);
	double t_2 = pow(((sin((0.5 * phi1)) * cos((phi2 * 0.5))) - sin((0.5 * phi2))), 2.0) + (cos(phi1) * (cos(phi2) * pow(fma(cos((-0.5 * lambda2)), sin((0.5 * lambda1)), (cos((0.5 * lambda1)) * sin((-0.5 * lambda2)))), 2.0)));
	double t_3 = R * (2.0 * atan2(sqrt(t_2), sqrt((1.0 - t_2))));
	double tmp;
	if (lambda1 <= -1.35e+34) {
		tmp = t_3;
	} else if (lambda1 <= 130000000.0) {
		tmp = R * (2.0 * atan2(sqrt(t_1), sqrt((1.0 - t_1))));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_1 = Float64((fma(sin(Float64(phi1 * 0.5)), cos(Float64(phi2 / -2.0)), Float64(cos(Float64(phi1 * 0.5)) * sin(Float64(phi2 / -2.0)))) ^ 2.0) + Float64(Float64(Float64(cos(phi1) * cos(phi2)) * t_0) * t_0))
	t_2 = Float64((Float64(Float64(sin(Float64(0.5 * phi1)) * cos(Float64(phi2 * 0.5))) - sin(Float64(0.5 * phi2))) ^ 2.0) + Float64(cos(phi1) * Float64(cos(phi2) * (fma(cos(Float64(-0.5 * lambda2)), sin(Float64(0.5 * lambda1)), Float64(cos(Float64(0.5 * lambda1)) * sin(Float64(-0.5 * lambda2)))) ^ 2.0))))
	t_3 = Float64(R * Float64(2.0 * atan(sqrt(t_2), sqrt(Float64(1.0 - t_2)))))
	tmp = 0.0
	if (lambda1 <= -1.35e+34)
		tmp = t_3;
	elseif (lambda1 <= 130000000.0)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_1), sqrt(Float64(1.0 - t_1)))));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[N[(N[Sin[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(phi2 / -2.0), $MachinePrecision]], $MachinePrecision] + N[(N[Cos[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(phi2 / -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[N[(N[(N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[Sin[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Power[N[(N[Cos[N[(-0.5 * lambda2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision] + N[(N[Cos[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(-0.5 * lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$2], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$2), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[lambda1, -1.35e+34], t$95$3, If[LessEqual[lambda1, 130000000.0], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$1], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]

Derivation

1. Split input into 2 regimes

2. if lambda1 < -1.35e34 or 1.3e8 < lambda1

   1. Initial program 62.0%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 61.1%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 62.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 62.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}}\right) \]

   9. Applied rewrites 77.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right)}}\right) \]
   10. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       7. mult-flip-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       8. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       9. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       10. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       11. lift-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       12. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       13. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       14. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       15. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       17. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       18. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       19. mult-flip-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       20. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       21. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       22. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       23. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       24. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       25. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   11. Applied rewrites 78.2%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   12. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       7. mult-flip-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       8. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       9. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       10. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       11. lift-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       12. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       13. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       14. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       15. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       17. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       18. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       19. mult-flip-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       20. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       21. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       22. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       23. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       24. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       25. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   13. Applied rewrites 98.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   14. Taylor expanded in lambda1 around inf

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   15. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \color{blue}{\left(\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\color{blue}{\cos \phi_2} \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \color{blue}{{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\color{blue}{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{\color{blue}{2}}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   16. Applied rewrites 98.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   17. Taylor expanded in lambda1 around inf

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}\right)}\right)}}\right) \]
   18. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \color{blue}{\left(\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}\right)}\right)}}\right) \]

       2. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\color{blue}{\cos \phi_2} \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}\right)\right)}}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \color{blue}{{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}\right)\right)}}\right) \]

       4. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\color{blue}{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}}^{2}\right)\right)}}\right) \]

       5. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{\color{blue}{2}}\right)\right)}}\right) \]

   19. Applied rewrites 98.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}\right)}\right)}}\right) \]

   20. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\sin \left(\frac{1}{2} \cdot \phi_2\right)}\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]
   21. Step-by-step derivation

       1. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       2. lower-*.f64 79.0

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \sin \left(0.5 \cdot \phi_2\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

   22. Applied rewrites 79.0%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \color{blue}{\sin \left(0.5 \cdot \phi_2\right)}\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

   23. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\sin \left(\frac{1}{2} \cdot \phi_2\right)}\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]
   24. Step-by-step derivation

       1. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       2. lower-*.f64 76.9

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \sin \left(0.5 \cdot \phi_2\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \sin \left(0.5 \cdot \phi_2\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

   25. Applied rewrites 76.9%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \sin \left(0.5 \cdot \phi_2\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \color{blue}{\sin \left(0.5 \cdot \phi_2\right)}\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

   if -1.35e34 < lambda1 < 1.3e8

   1. Initial program 62.0%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 62.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 77.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 86.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := {\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_0\right) \cdot t\_0\\ t_2 := \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right) + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}\right)\\ t_3 := R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_2}}{\sqrt{1 - t\_2}}\right)\\ \mathbf{if}\;\lambda_1 \leq -1.35 \cdot 10^{+34}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;\lambda_1 \leq 130000000:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_1}}{\sqrt{1 - t\_1}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_1
         (+
          (pow
           (fma
            (sin (* phi1 0.5))
            (cos (/ phi2 -2.0))
            (* (cos (* phi1 0.5)) (sin (/ phi2 -2.0))))
           2.0)
          (* (* (* (cos phi1) (cos phi2)) t_0) t_0)))
        (t_2
         (+
          (- 0.5 (* 0.5 (cos (* 2.0 (* (- phi1 phi2) 0.5)))))
          (*
           (cos phi1)
           (*
            (cos phi2)
            (pow
             (fma
              (cos (* -0.5 lambda2))
              (sin (* 0.5 lambda1))
              (* (cos (* 0.5 lambda1)) (sin (* -0.5 lambda2))))
             2.0)))))
        (t_3 (* R (* 2.0 (atan2 (sqrt t_2) (sqrt (- 1.0 t_2)))))))
   (if (<= lambda1 -1.35e+34)
     t_3
     (if (<= lambda1 130000000.0)
       (* R (* 2.0 (atan2 (sqrt t_1) (sqrt (- 1.0 t_1)))))
       t_3))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(((lambda1 - lambda2) / 2.0));
	double t_1 = pow(fma(sin((phi1 * 0.5)), cos((phi2 / -2.0)), (cos((phi1 * 0.5)) * sin((phi2 / -2.0)))), 2.0) + (((cos(phi1) * cos(phi2)) * t_0) * t_0);
	double t_2 = (0.5 - (0.5 * cos((2.0 * ((phi1 - phi2) * 0.5))))) + (cos(phi1) * (cos(phi2) * pow(fma(cos((-0.5 * lambda2)), sin((0.5 * lambda1)), (cos((0.5 * lambda1)) * sin((-0.5 * lambda2)))), 2.0)));
	double t_3 = R * (2.0 * atan2(sqrt(t_2), sqrt((1.0 - t_2))));
	double tmp;
	if (lambda1 <= -1.35e+34) {
		tmp = t_3;
	} else if (lambda1 <= 130000000.0) {
		tmp = R * (2.0 * atan2(sqrt(t_1), sqrt((1.0 - t_1))));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_1 = Float64((fma(sin(Float64(phi1 * 0.5)), cos(Float64(phi2 / -2.0)), Float64(cos(Float64(phi1 * 0.5)) * sin(Float64(phi2 / -2.0)))) ^ 2.0) + Float64(Float64(Float64(cos(phi1) * cos(phi2)) * t_0) * t_0))
	t_2 = Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(Float64(phi1 - phi2) * 0.5))))) + Float64(cos(phi1) * Float64(cos(phi2) * (fma(cos(Float64(-0.5 * lambda2)), sin(Float64(0.5 * lambda1)), Float64(cos(Float64(0.5 * lambda1)) * sin(Float64(-0.5 * lambda2)))) ^ 2.0))))
	t_3 = Float64(R * Float64(2.0 * atan(sqrt(t_2), sqrt(Float64(1.0 - t_2)))))
	tmp = 0.0
	if (lambda1 <= -1.35e+34)
		tmp = t_3;
	elseif (lambda1 <= 130000000.0)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_1), sqrt(Float64(1.0 - t_1)))));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[N[(N[Sin[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(phi2 / -2.0), $MachinePrecision]], $MachinePrecision] + N[(N[Cos[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(phi2 / -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(N[(phi1 - phi2), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Power[N[(N[Cos[N[(-0.5 * lambda2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision] + N[(N[Cos[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(-0.5 * lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$2], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$2), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[lambda1, -1.35e+34], t$95$3, If[LessEqual[lambda1, 130000000.0], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$1], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]

Derivation

1. Split input into 2 regimes

2. if lambda1 < -1.35e34 or 1.3e8 < lambda1

   1. Initial program 62.0%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 61.1%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 62.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 62.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}}\right) \]

   9. Applied rewrites 77.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right)}}\right) \]
   10. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       7. mult-flip-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       8. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       9. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       10. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       11. lift-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       12. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       13. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       14. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       15. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       17. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       18. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       19. mult-flip-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       20. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       21. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       22. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       23. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       24. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       25. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   11. Applied rewrites 78.2%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   12. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       7. mult-flip-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       8. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       9. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       10. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       11. lift-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       12. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       13. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       14. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       15. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       17. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       18. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       19. mult-flip-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       20. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       21. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       22. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       23. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       24. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       25. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   13. Applied rewrites 98.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   14. Taylor expanded in lambda1 around inf

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   15. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \color{blue}{\left(\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\color{blue}{\cos \phi_2} \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \color{blue}{{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\color{blue}{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{\color{blue}{2}}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   16. Applied rewrites 98.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   17. Taylor expanded in lambda1 around inf

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}\right)}\right)}}\right) \]
   18. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \color{blue}{\left(\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}\right)}\right)}}\right) \]

       2. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\color{blue}{\cos \phi_2} \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}\right)\right)}}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \color{blue}{{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}\right)\right)}}\right) \]

       4. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\color{blue}{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}}^{2}\right)\right)}}\right) \]

       5. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{\color{blue}{2}}\right)\right)}}\right) \]

   19. Applied rewrites 98.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}\right)}\right)}}\right) \]
   20. Step-by-step derivation

       1. lift-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2}} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       2. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       3. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       4. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       5. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\cos \left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       6. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)}\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       7. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       8. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\sin \left(\phi_2 \cdot \frac{1}{2}\right)}\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       9. sin-diff-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1 - \phi_2 \cdot \frac{1}{2}\right)}}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       10. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\color{blue}{\frac{1}{2} \cdot \phi_1} - \phi_2 \cdot \frac{1}{2}\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       11. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\color{blue}{\phi_1 \cdot \frac{1}{2}} - \phi_2 \cdot \frac{1}{2}\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       12. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\phi_1 \cdot \frac{1}{2} - \color{blue}{\phi_2 \cdot \frac{1}{2}}\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       13. distribute-rgt-out-- N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       14. lift--.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \color{blue}{\left(\phi_1 - \phi_2\right)}\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       15. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)}}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       16. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\left(\phi_1 - \phi_2\right) \cdot \color{blue}{\frac{1}{2}}\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       17. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       18. lift-/.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       19. lift-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

   21. Applied rewrites 75.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]
   22. Step-by-step derivation

       1. lift-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left(\color{blue}{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2}} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       2. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       3. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       4. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       5. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\cos \left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       6. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)}\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       7. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       8. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\sin \left(\phi_2 \cdot \frac{1}{2}\right)}\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       9. sin-diff-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1 - \phi_2 \cdot \frac{1}{2}\right)}}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       10. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\color{blue}{\frac{1}{2} \cdot \phi_1} - \phi_2 \cdot \frac{1}{2}\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       11. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\color{blue}{\phi_1 \cdot \frac{1}{2}} - \phi_2 \cdot \frac{1}{2}\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       12. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\phi_1 \cdot \frac{1}{2} - \color{blue}{\phi_2 \cdot \frac{1}{2}}\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       13. distribute-rgt-out-- N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       14. lift--.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{1}{2} \cdot \color{blue}{\left(\phi_1 - \phi_2\right)}\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       15. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)}}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       16. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\left(\phi_1 - \phi_2\right) \cdot \color{blue}{\frac{1}{2}}\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       17. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       18. lift-/.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       19. lift-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

   23. Applied rewrites 74.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right) + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left(\color{blue}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

   if -1.35e34 < lambda1 < 1.3e8

   1. Initial program 62.0%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 62.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 77.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 86.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := {\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_0\right) \cdot t\_0\\ t_2 := \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right) + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}\right)\\ t_3 := R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_2}}{\sqrt{1 - t\_2}}\right)\\ \mathbf{if}\;\lambda_1 \leq -1.35 \cdot 10^{+34}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;\lambda_1 \leq 130000000:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_1}}{\sqrt{1 - t\_1}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_1
         (+
          (pow
           (-
            (* (sin (* phi1 0.5)) (cos (* phi2 0.5)))
            (* (cos (* phi1 0.5)) (sin (* phi2 0.5))))
           2.0)
          (* (* (* (cos phi1) (cos phi2)) t_0) t_0)))
        (t_2
         (+
          (- 0.5 (* 0.5 (cos (* 2.0 (* (- phi1 phi2) 0.5)))))
          (*
           (cos phi1)
           (*
            (cos phi2)
            (pow
             (fma
              (cos (* -0.5 lambda2))
              (sin (* 0.5 lambda1))
              (* (cos (* 0.5 lambda1)) (sin (* -0.5 lambda2))))
             2.0)))))
        (t_3 (* R (* 2.0 (atan2 (sqrt t_2) (sqrt (- 1.0 t_2)))))))
   (if (<= lambda1 -1.35e+34)
     t_3
     (if (<= lambda1 130000000.0)
       (* R (* 2.0 (atan2 (sqrt t_1) (sqrt (- 1.0 t_1)))))
       t_3))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(((lambda1 - lambda2) / 2.0));
	double t_1 = pow(((sin((phi1 * 0.5)) * cos((phi2 * 0.5))) - (cos((phi1 * 0.5)) * sin((phi2 * 0.5)))), 2.0) + (((cos(phi1) * cos(phi2)) * t_0) * t_0);
	double t_2 = (0.5 - (0.5 * cos((2.0 * ((phi1 - phi2) * 0.5))))) + (cos(phi1) * (cos(phi2) * pow(fma(cos((-0.5 * lambda2)), sin((0.5 * lambda1)), (cos((0.5 * lambda1)) * sin((-0.5 * lambda2)))), 2.0)));
	double t_3 = R * (2.0 * atan2(sqrt(t_2), sqrt((1.0 - t_2))));
	double tmp;
	if (lambda1 <= -1.35e+34) {
		tmp = t_3;
	} else if (lambda1 <= 130000000.0) {
		tmp = R * (2.0 * atan2(sqrt(t_1), sqrt((1.0 - t_1))));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_1 = Float64((Float64(Float64(sin(Float64(phi1 * 0.5)) * cos(Float64(phi2 * 0.5))) - Float64(cos(Float64(phi1 * 0.5)) * sin(Float64(phi2 * 0.5)))) ^ 2.0) + Float64(Float64(Float64(cos(phi1) * cos(phi2)) * t_0) * t_0))
	t_2 = Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(Float64(phi1 - phi2) * 0.5))))) + Float64(cos(phi1) * Float64(cos(phi2) * (fma(cos(Float64(-0.5 * lambda2)), sin(Float64(0.5 * lambda1)), Float64(cos(Float64(0.5 * lambda1)) * sin(Float64(-0.5 * lambda2)))) ^ 2.0))))
	t_3 = Float64(R * Float64(2.0 * atan(sqrt(t_2), sqrt(Float64(1.0 - t_2)))))
	tmp = 0.0
	if (lambda1 <= -1.35e+34)
		tmp = t_3;
	elseif (lambda1 <= 130000000.0)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_1), sqrt(Float64(1.0 - t_1)))));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[N[(N[(N[Sin[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(N[(phi1 - phi2), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Power[N[(N[Cos[N[(-0.5 * lambda2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision] + N[(N[Cos[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(-0.5 * lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$2], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$2), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[lambda1, -1.35e+34], t$95$3, If[LessEqual[lambda1, 130000000.0], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$1], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]

Derivation

1. Split input into 2 regimes

2. if lambda1 < -1.35e34 or 1.3e8 < lambda1

   1. Initial program 62.0%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 61.1%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 62.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 62.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}}\right) \]

   9. Applied rewrites 77.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right)}}\right) \]
   10. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       7. mult-flip-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       8. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       9. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       10. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       11. lift-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       12. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       13. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       14. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       15. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       17. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       18. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       19. mult-flip-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       20. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       21. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       22. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       23. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       24. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       25. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   11. Applied rewrites 78.2%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   12. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       7. mult-flip-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       8. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       9. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       10. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       11. lift-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       12. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       13. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       14. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       15. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       17. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       18. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       19. mult-flip-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       20. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       21. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       22. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       23. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       24. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       25. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   13. Applied rewrites 98.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   14. Taylor expanded in lambda1 around inf

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   15. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \color{blue}{\left(\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\color{blue}{\cos \phi_2} \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \color{blue}{{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\color{blue}{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{\color{blue}{2}}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   16. Applied rewrites 98.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   17. Taylor expanded in lambda1 around inf

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}\right)}\right)}}\right) \]
   18. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \color{blue}{\left(\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}\right)}\right)}}\right) \]

       2. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\color{blue}{\cos \phi_2} \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}\right)\right)}}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \color{blue}{{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}\right)\right)}}\right) \]

       4. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\color{blue}{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}}^{2}\right)\right)}}\right) \]

       5. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{\color{blue}{2}}\right)\right)}}\right) \]

   19. Applied rewrites 98.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}\right)}\right)}}\right) \]
   20. Step-by-step derivation

       1. lift-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2}} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       2. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       3. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       4. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       5. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\cos \left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       6. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)}\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       7. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       8. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\sin \left(\phi_2 \cdot \frac{1}{2}\right)}\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       9. sin-diff-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1 - \phi_2 \cdot \frac{1}{2}\right)}}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       10. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\color{blue}{\frac{1}{2} \cdot \phi_1} - \phi_2 \cdot \frac{1}{2}\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       11. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\color{blue}{\phi_1 \cdot \frac{1}{2}} - \phi_2 \cdot \frac{1}{2}\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       12. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\phi_1 \cdot \frac{1}{2} - \color{blue}{\phi_2 \cdot \frac{1}{2}}\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       13. distribute-rgt-out-- N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       14. lift--.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \color{blue}{\left(\phi_1 - \phi_2\right)}\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       15. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)}}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       16. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\left(\phi_1 - \phi_2\right) \cdot \color{blue}{\frac{1}{2}}\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       17. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       18. lift-/.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       19. lift-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

   21. Applied rewrites 75.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]
   22. Step-by-step derivation

       1. lift-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left(\color{blue}{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2}} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       2. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       3. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       4. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       5. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\cos \left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       6. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)}\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       7. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       8. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\sin \left(\phi_2 \cdot \frac{1}{2}\right)}\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       9. sin-diff-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1 - \phi_2 \cdot \frac{1}{2}\right)}}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       10. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\color{blue}{\frac{1}{2} \cdot \phi_1} - \phi_2 \cdot \frac{1}{2}\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       11. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\color{blue}{\phi_1 \cdot \frac{1}{2}} - \phi_2 \cdot \frac{1}{2}\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       12. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\phi_1 \cdot \frac{1}{2} - \color{blue}{\phi_2 \cdot \frac{1}{2}}\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       13. distribute-rgt-out-- N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       14. lift--.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{1}{2} \cdot \color{blue}{\left(\phi_1 - \phi_2\right)}\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       15. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)}}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       16. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\left(\phi_1 - \phi_2\right) \cdot \color{blue}{\frac{1}{2}}\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       17. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       18. lift-/.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       19. lift-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

   23. Applied rewrites 74.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right) + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left(\color{blue}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

   if -1.35e34 < lambda1 < 1.3e8

   1. Initial program 62.0%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sin-diff N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. lower-*.f64 62.9

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\phi_2 \cdot 0.5\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 62.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sin-diff N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. lower-*.f64 77.9

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\phi_2 \cdot 0.5\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 77.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 86.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_0 := \sin \left(-0.5 \cdot \lambda_2\right)\\ t_1 := {\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {t\_0}^{2}\right)\\ t_2 := \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right) + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot t\_0\right)\right)}^{2}\right)\\ t_3 := R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_2}}{\sqrt{1 - t\_2}}\right)\\ \mathbf{if}\;\lambda_1 \leq -2.2 \cdot 10^{-10}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;\lambda_1 \leq 8.6 \cdot 10^{-9}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_1}}{\sqrt{1 - t\_1}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (* -0.5 lambda2)))
        (t_1
         (+
          (pow
           (-
            (* (sin (* 0.5 phi1)) (cos (* phi2 0.5)))
            (* (cos (* 0.5 phi1)) (sin (* phi2 0.5))))
           2.0)
          (* (cos phi1) (* (cos phi2) (pow t_0 2.0)))))
        (t_2
         (+
          (- 0.5 (* 0.5 (cos (* 2.0 (* (- phi1 phi2) 0.5)))))
          (*
           (cos phi1)
           (*
            (cos phi2)
            (pow
             (fma
              (cos (* -0.5 lambda2))
              (sin (* 0.5 lambda1))
              (* (cos (* 0.5 lambda1)) t_0))
             2.0)))))
        (t_3 (* R (* 2.0 (atan2 (sqrt t_2) (sqrt (- 1.0 t_2)))))))
   (if (<= lambda1 -2.2e-10)
     t_3
     (if (<= lambda1 8.6e-9)
       (* R (* 2.0 (atan2 (sqrt t_1) (sqrt (- 1.0 t_1)))))
       t_3))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin((-0.5 * lambda2));
	double t_1 = pow(((sin((0.5 * phi1)) * cos((phi2 * 0.5))) - (cos((0.5 * phi1)) * sin((phi2 * 0.5)))), 2.0) + (cos(phi1) * (cos(phi2) * pow(t_0, 2.0)));
	double t_2 = (0.5 - (0.5 * cos((2.0 * ((phi1 - phi2) * 0.5))))) + (cos(phi1) * (cos(phi2) * pow(fma(cos((-0.5 * lambda2)), sin((0.5 * lambda1)), (cos((0.5 * lambda1)) * t_0)), 2.0)));
	double t_3 = R * (2.0 * atan2(sqrt(t_2), sqrt((1.0 - t_2))));
	double tmp;
	if (lambda1 <= -2.2e-10) {
		tmp = t_3;
	} else if (lambda1 <= 8.6e-9) {
		tmp = R * (2.0 * atan2(sqrt(t_1), sqrt((1.0 - t_1))));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(-0.5 * lambda2))
	t_1 = Float64((Float64(Float64(sin(Float64(0.5 * phi1)) * cos(Float64(phi2 * 0.5))) - Float64(cos(Float64(0.5 * phi1)) * sin(Float64(phi2 * 0.5)))) ^ 2.0) + Float64(cos(phi1) * Float64(cos(phi2) * (t_0 ^ 2.0))))
	t_2 = Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(Float64(phi1 - phi2) * 0.5))))) + Float64(cos(phi1) * Float64(cos(phi2) * (fma(cos(Float64(-0.5 * lambda2)), sin(Float64(0.5 * lambda1)), Float64(cos(Float64(0.5 * lambda1)) * t_0)) ^ 2.0))))
	t_3 = Float64(R * Float64(2.0 * atan(sqrt(t_2), sqrt(Float64(1.0 - t_2)))))
	tmp = 0.0
	if (lambda1 <= -2.2e-10)
		tmp = t_3;
	elseif (lambda1 <= 8.6e-9)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_1), sqrt(Float64(1.0 - t_1)))));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(-0.5 * lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[N[(N[(N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(N[(phi1 - phi2), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Power[N[(N[Cos[N[(-0.5 * lambda2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision] + N[(N[Cos[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$2], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$2), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[lambda1, -2.2e-10], t$95$3, If[LessEqual[lambda1, 8.6e-9], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$1], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]

Derivation

1. Split input into 2 regimes

2. if lambda1 < -2.1999999999999999e-10 or 8.59999999999999925e-9 < lambda1

   1. Initial program 62.0%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 61.1%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 62.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 62.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}}\right) \]

   9. Applied rewrites 77.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right)}}\right) \]
   10. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       7. mult-flip-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       8. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       9. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       10. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       11. lift-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       12. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       13. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       14. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       15. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       17. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       18. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       19. mult-flip-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       20. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       21. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       22. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       23. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       24. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       25. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   11. Applied rewrites 78.2%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   12. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       7. mult-flip-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       8. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       9. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       10. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       11. lift-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       12. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       13. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       14. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       15. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       17. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       18. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       19. mult-flip-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       20. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       21. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       22. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       23. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       24. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       25. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   13. Applied rewrites 98.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   14. Taylor expanded in lambda1 around inf

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   15. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \color{blue}{\left(\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\color{blue}{\cos \phi_2} \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \color{blue}{{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\color{blue}{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{\color{blue}{2}}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   16. Applied rewrites 98.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   17. Taylor expanded in lambda1 around inf

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}\right)}\right)}}\right) \]
   18. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \color{blue}{\left(\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}\right)}\right)}}\right) \]

       2. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\color{blue}{\cos \phi_2} \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}\right)\right)}}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \color{blue}{{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}\right)\right)}}\right) \]

       4. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\color{blue}{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}}^{2}\right)\right)}}\right) \]

       5. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{\color{blue}{2}}\right)\right)}}\right) \]

   19. Applied rewrites 98.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}\right)}\right)}}\right) \]
   20. Step-by-step derivation

       1. lift-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2}} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       2. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       3. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       4. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       5. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\cos \left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       6. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)}\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       7. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       8. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\sin \left(\phi_2 \cdot \frac{1}{2}\right)}\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       9. sin-diff-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1 - \phi_2 \cdot \frac{1}{2}\right)}}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       10. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\color{blue}{\frac{1}{2} \cdot \phi_1} - \phi_2 \cdot \frac{1}{2}\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       11. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\color{blue}{\phi_1 \cdot \frac{1}{2}} - \phi_2 \cdot \frac{1}{2}\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       12. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\phi_1 \cdot \frac{1}{2} - \color{blue}{\phi_2 \cdot \frac{1}{2}}\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       13. distribute-rgt-out-- N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       14. lift--.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \color{blue}{\left(\phi_1 - \phi_2\right)}\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       15. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)}}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       16. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\left(\phi_1 - \phi_2\right) \cdot \color{blue}{\frac{1}{2}}\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       17. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       18. lift-/.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       19. lift-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

   21. Applied rewrites 75.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]
   22. Step-by-step derivation

       1. lift-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left(\color{blue}{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2}} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       2. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       3. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       4. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       5. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\cos \left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       6. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)}\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       7. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       8. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\sin \left(\phi_2 \cdot \frac{1}{2}\right)}\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       9. sin-diff-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1 - \phi_2 \cdot \frac{1}{2}\right)}}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       10. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\color{blue}{\frac{1}{2} \cdot \phi_1} - \phi_2 \cdot \frac{1}{2}\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       11. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\color{blue}{\phi_1 \cdot \frac{1}{2}} - \phi_2 \cdot \frac{1}{2}\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       12. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\phi_1 \cdot \frac{1}{2} - \color{blue}{\phi_2 \cdot \frac{1}{2}}\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       13. distribute-rgt-out-- N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       14. lift--.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{1}{2} \cdot \color{blue}{\left(\phi_1 - \phi_2\right)}\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       15. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)}}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       16. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\left(\phi_1 - \phi_2\right) \cdot \color{blue}{\frac{1}{2}}\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       17. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       18. lift-/.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       19. lift-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

   23. Applied rewrites 74.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right) + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left(\color{blue}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

   if -2.1999999999999999e-10 < lambda1 < 8.59999999999999925e-9

   1. Initial program 62.0%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 61.1%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 62.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 62.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}}\right) \]

   9. Applied rewrites 77.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right)}}\right) \]
   10. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       7. mult-flip-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       8. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       9. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       10. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       11. lift-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       12. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       13. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       14. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       15. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       17. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       18. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       19. mult-flip-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       20. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       21. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       22. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       23. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       24. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       25. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   11. Applied rewrites 78.2%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   12. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       7. mult-flip-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       8. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       9. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       10. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       11. lift-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       12. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       13. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       14. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       15. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       17. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       18. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       19. mult-flip-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       20. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       21. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       22. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       23. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       24. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       25. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   13. Applied rewrites 98.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   14. Taylor expanded in lambda1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   15. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \color{blue}{\left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\color{blue}{\cos \phi_2} \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \color{blue}{{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\color{blue}{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{\color{blue}{2}}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       6. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       7. lower-*.f64 57.5

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   16. Applied rewrites 57.5%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   17. Taylor expanded in lambda1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}\right)}\right)}}\right) \]
   18. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \color{blue}{\left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}\right)}\right)}}\right) \]

       2. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\color{blue}{\cos \phi_2} \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}\right)\right)}}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \color{blue}{{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}\right)\right)}}\right) \]

       4. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\color{blue}{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}}^{2}\right)\right)}}\right) \]

       5. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{\color{blue}{2}}\right)\right)}}\right) \]

       6. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}\right)\right)}}\right) \]

       7. lower-*.f64 56.6

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}\right)\right)}}\right) \]

   19. Applied rewrites 56.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}\right)}\right)}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 86.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_0 := \sin \left(-0.5 \cdot \lambda_2\right)\\ t_1 := \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {t\_0}^{2}, {\left(\cos \left(0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\right)\\ t_2 := \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right) + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot t\_0\right)\right)}^{2}\right)\\ t_3 := R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_2}}{\sqrt{1 - t\_2}}\right)\\ \mathbf{if}\;\lambda_1 \leq -2.2 \cdot 10^{-10}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;\lambda_1 \leq 8.6 \cdot 10^{-9}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_1}}{\sqrt{1 - t\_1}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (* -0.5 lambda2)))
        (t_1
         (fma
          (cos phi1)
          (* (cos phi2) (pow t_0 2.0))
          (pow
           (-
            (* (cos (* 0.5 phi2)) (sin (* 0.5 phi1)))
            (* (cos (* 0.5 phi1)) (sin (* 0.5 phi2))))
           2.0)))
        (t_2
         (+
          (- 0.5 (* 0.5 (cos (* 2.0 (* (- phi1 phi2) 0.5)))))
          (*
           (cos phi1)
           (*
            (cos phi2)
            (pow
             (fma
              (cos (* -0.5 lambda2))
              (sin (* 0.5 lambda1))
              (* (cos (* 0.5 lambda1)) t_0))
             2.0)))))
        (t_3 (* R (* 2.0 (atan2 (sqrt t_2) (sqrt (- 1.0 t_2)))))))
   (if (<= lambda1 -2.2e-10)
     t_3
     (if (<= lambda1 8.6e-9)
       (* R (* 2.0 (atan2 (sqrt t_1) (sqrt (- 1.0 t_1)))))
       t_3))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin((-0.5 * lambda2));
	double t_1 = fma(cos(phi1), (cos(phi2) * pow(t_0, 2.0)), pow(((cos((0.5 * phi2)) * sin((0.5 * phi1))) - (cos((0.5 * phi1)) * sin((0.5 * phi2)))), 2.0));
	double t_2 = (0.5 - (0.5 * cos((2.0 * ((phi1 - phi2) * 0.5))))) + (cos(phi1) * (cos(phi2) * pow(fma(cos((-0.5 * lambda2)), sin((0.5 * lambda1)), (cos((0.5 * lambda1)) * t_0)), 2.0)));
	double t_3 = R * (2.0 * atan2(sqrt(t_2), sqrt((1.0 - t_2))));
	double tmp;
	if (lambda1 <= -2.2e-10) {
		tmp = t_3;
	} else if (lambda1 <= 8.6e-9) {
		tmp = R * (2.0 * atan2(sqrt(t_1), sqrt((1.0 - t_1))));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(-0.5 * lambda2))
	t_1 = fma(cos(phi1), Float64(cos(phi2) * (t_0 ^ 2.0)), (Float64(Float64(cos(Float64(0.5 * phi2)) * sin(Float64(0.5 * phi1))) - Float64(cos(Float64(0.5 * phi1)) * sin(Float64(0.5 * phi2)))) ^ 2.0))
	t_2 = Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(Float64(phi1 - phi2) * 0.5))))) + Float64(cos(phi1) * Float64(cos(phi2) * (fma(cos(Float64(-0.5 * lambda2)), sin(Float64(0.5 * lambda1)), Float64(cos(Float64(0.5 * lambda1)) * t_0)) ^ 2.0))))
	t_3 = Float64(R * Float64(2.0 * atan(sqrt(t_2), sqrt(Float64(1.0 - t_2)))))
	tmp = 0.0
	if (lambda1 <= -2.2e-10)
		tmp = t_3;
	elseif (lambda1 <= 8.6e-9)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_1), sqrt(Float64(1.0 - t_1)))));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(-0.5 * lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision] + N[Power[N[(N[(N[Cos[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(N[(phi1 - phi2), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Power[N[(N[Cos[N[(-0.5 * lambda2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision] + N[(N[Cos[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$2], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$2), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[lambda1, -2.2e-10], t$95$3, If[LessEqual[lambda1, 8.6e-9], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$1], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]

Derivation

1. Split input into 2 regimes

2. if lambda1 < -2.1999999999999999e-10 or 8.59999999999999925e-9 < lambda1

   1. Initial program 62.0%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 61.1%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 62.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 62.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}}\right) \]

   9. Applied rewrites 77.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right)}}\right) \]
   10. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       7. mult-flip-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       8. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       9. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       10. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       11. lift-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       12. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       13. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       14. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       15. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       17. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       18. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       19. mult-flip-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       20. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       21. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       22. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       23. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       24. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       25. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   11. Applied rewrites 78.2%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   12. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       7. mult-flip-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       8. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       9. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       10. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       11. lift-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       12. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       13. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       14. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       15. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       17. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       18. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       19. mult-flip-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       20. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       21. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       22. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       23. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       24. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       25. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   13. Applied rewrites 98.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   14. Taylor expanded in lambda1 around inf

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   15. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \color{blue}{\left(\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\color{blue}{\cos \phi_2} \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \color{blue}{{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\color{blue}{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{\color{blue}{2}}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   16. Applied rewrites 98.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   17. Taylor expanded in lambda1 around inf

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}\right)}\right)}}\right) \]
   18. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \color{blue}{\left(\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}\right)}\right)}}\right) \]

       2. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\color{blue}{\cos \phi_2} \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}\right)\right)}}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \color{blue}{{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}\right)\right)}}\right) \]

       4. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\color{blue}{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}}^{2}\right)\right)}}\right) \]

       5. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{\color{blue}{2}}\right)\right)}}\right) \]

   19. Applied rewrites 98.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}\right)}\right)}}\right) \]
   20. Step-by-step derivation

       1. lift-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2}} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       2. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       3. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       4. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       5. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\cos \left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       6. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)}\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       7. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       8. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\sin \left(\phi_2 \cdot \frac{1}{2}\right)}\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       9. sin-diff-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1 - \phi_2 \cdot \frac{1}{2}\right)}}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       10. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\color{blue}{\frac{1}{2} \cdot \phi_1} - \phi_2 \cdot \frac{1}{2}\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       11. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\color{blue}{\phi_1 \cdot \frac{1}{2}} - \phi_2 \cdot \frac{1}{2}\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       12. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\phi_1 \cdot \frac{1}{2} - \color{blue}{\phi_2 \cdot \frac{1}{2}}\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       13. distribute-rgt-out-- N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       14. lift--.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \color{blue}{\left(\phi_1 - \phi_2\right)}\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       15. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)}}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       16. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\left(\phi_1 - \phi_2\right) \cdot \color{blue}{\frac{1}{2}}\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       17. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       18. lift-/.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       19. lift-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

   21. Applied rewrites 75.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]
   22. Step-by-step derivation

       1. lift-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left(\color{blue}{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2}} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       2. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       3. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       4. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       5. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\cos \left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       6. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)}\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       7. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       8. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\sin \left(\phi_2 \cdot \frac{1}{2}\right)}\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       9. sin-diff-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1 - \phi_2 \cdot \frac{1}{2}\right)}}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       10. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\color{blue}{\frac{1}{2} \cdot \phi_1} - \phi_2 \cdot \frac{1}{2}\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       11. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\color{blue}{\phi_1 \cdot \frac{1}{2}} - \phi_2 \cdot \frac{1}{2}\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       12. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\phi_1 \cdot \frac{1}{2} - \color{blue}{\phi_2 \cdot \frac{1}{2}}\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       13. distribute-rgt-out-- N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       14. lift--.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{1}{2} \cdot \color{blue}{\left(\phi_1 - \phi_2\right)}\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       15. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)}}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       16. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\left(\phi_1 - \phi_2\right) \cdot \color{blue}{\frac{1}{2}}\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       17. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       18. lift-/.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

       19. lift-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right) + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

   23. Applied rewrites 74.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right) + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left(\color{blue}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}\right)\right)}}\right) \]

   if -2.1999999999999999e-10 < lambda1 < 8.59999999999999925e-9

   1. Initial program 62.0%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 61.1%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 62.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 62.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}}\right) \]

   9. Applied rewrites 77.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right)}}\right) \]
   10. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       7. mult-flip-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       8. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       9. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       10. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       11. lift-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       12. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       13. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       14. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       15. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       17. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       18. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       19. mult-flip-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       20. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       21. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       22. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       23. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       24. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       25. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   11. Applied rewrites 78.2%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   12. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       7. mult-flip-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       8. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       9. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       10. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       11. lift-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       12. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       13. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       14. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       15. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       17. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       18. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       19. mult-flip-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       20. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       21. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       22. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       23. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       24. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       25. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   13. Applied rewrites 98.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   14. Taylor expanded in lambda1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}\right) + {\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   15. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   16. Applied rewrites 57.5%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\left(\cos \left(0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   17. Taylor expanded in lambda1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}\right) + {\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}}\right) \]
   18. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

   19. Applied rewrites 56.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\left(\cos \left(0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\left(\cos \left(0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\right)}}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 86.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_0 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}\\ t_1 := {\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)\\ t_2 := R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_1}}{\sqrt{1 - t\_1}}\right)\\ \mathbf{if}\;\phi_2 \leq -1650000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\phi_2 \leq 10^{-20}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_0}}{\sqrt{1 - t\_0}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (+
          (pow (sin (/ (- phi1 phi2) 2.0)) 2.0)
          (*
           (cos phi1)
           (pow
            (fma
             (cos (* -0.5 lambda2))
             (sin (* 0.5 lambda1))
             (* (cos (* 0.5 lambda1)) (sin (* -0.5 lambda2))))
            2.0))))
        (t_1
         (+
          (pow
           (-
            (* (sin (* 0.5 phi1)) (cos (* phi2 0.5)))
            (* (cos (* 0.5 phi1)) (sin (* phi2 0.5))))
           2.0)
          (*
           (- 0.5 (* (cos (* 1.0 (- lambda1 lambda2))) 0.5))
           (* (cos phi2) (cos phi1)))))
        (t_2 (* R (* 2.0 (atan2 (sqrt t_1) (sqrt (- 1.0 t_1)))))))
   (if (<= phi2 -1650000000.0)
     t_2
     (if (<= phi2 1e-20)
       (* R (* 2.0 (atan2 (sqrt t_0) (sqrt (- 1.0 t_0)))))
       t_2))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = pow(sin(((phi1 - phi2) / 2.0)), 2.0) + (cos(phi1) * pow(fma(cos((-0.5 * lambda2)), sin((0.5 * lambda1)), (cos((0.5 * lambda1)) * sin((-0.5 * lambda2)))), 2.0));
	double t_1 = pow(((sin((0.5 * phi1)) * cos((phi2 * 0.5))) - (cos((0.5 * phi1)) * sin((phi2 * 0.5)))), 2.0) + ((0.5 - (cos((1.0 * (lambda1 - lambda2))) * 0.5)) * (cos(phi2) * cos(phi1)));
	double t_2 = R * (2.0 * atan2(sqrt(t_1), sqrt((1.0 - t_1))));
	double tmp;
	if (phi2 <= -1650000000.0) {
		tmp = t_2;
	} else if (phi2 <= 1e-20) {
		tmp = R * (2.0 * atan2(sqrt(t_0), sqrt((1.0 - t_0))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64((sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0) + Float64(cos(phi1) * (fma(cos(Float64(-0.5 * lambda2)), sin(Float64(0.5 * lambda1)), Float64(cos(Float64(0.5 * lambda1)) * sin(Float64(-0.5 * lambda2)))) ^ 2.0)))
	t_1 = Float64((Float64(Float64(sin(Float64(0.5 * phi1)) * cos(Float64(phi2 * 0.5))) - Float64(cos(Float64(0.5 * phi1)) * sin(Float64(phi2 * 0.5)))) ^ 2.0) + Float64(Float64(0.5 - Float64(cos(Float64(1.0 * Float64(lambda1 - lambda2))) * 0.5)) * Float64(cos(phi2) * cos(phi1))))
	t_2 = Float64(R * Float64(2.0 * atan(sqrt(t_1), sqrt(Float64(1.0 - t_1)))))
	tmp = 0.0
	if (phi2 <= -1650000000.0)
		tmp = t_2;
	elseif (phi2 <= 1e-20)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_0), sqrt(Float64(1.0 - t_0)))));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * N[Power[N[(N[Cos[N[(-0.5 * lambda2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision] + N[(N[Cos[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(-0.5 * lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[N[(N[(N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(N[(0.5 - N[(N[Cos[N[(1.0 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$1], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -1650000000.0], t$95$2, If[LessEqual[phi2, 1e-20], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$0], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -1.65e9 or 9.99999999999999945e-21 < phi2

   1. Initial program 62.0%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 61.1%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 62.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 62.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}}\right) \]

   9. Applied rewrites 77.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right)}}\right) \]
   10. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       7. mult-flip-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       8. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       9. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       10. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       11. lift-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       12. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       13. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       14. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       15. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       17. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       18. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       19. mult-flip-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       20. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       21. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       22. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       23. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       24. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       25. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   11. Applied rewrites 78.2%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   12. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       7. mult-flip-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       8. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       9. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       10. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       11. lift-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       12. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       13. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       14. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       15. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       17. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       18. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       19. mult-flip-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       20. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       21. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       22. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       23. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       24. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       25. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   13. Applied rewrites 98.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   15. Applied rewrites 76.1%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \color{blue}{\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}}}{\sqrt{1 - \left({\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   17. Applied rewrites 75.5%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}}{\sqrt{1 - \left({\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \color{blue}{\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}\right)}}\right) \]

   if -1.65e9 < phi2 < 9.99999999999999945e-21

   1. Initial program 62.0%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 61.1%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 62.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 62.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}}\right) \]

   9. Applied rewrites 77.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right)}}\right) \]

   10. Taylor expanded in phi2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   11. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot \color{blue}{{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\color{blue}{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{\color{blue}{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   12. Applied rewrites 63.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   13. Taylor expanded in phi2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}\right)}}\right) \]
   14. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot \color{blue}{{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}\right)}}\right) \]

       2. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\color{blue}{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}}^{2}\right)}}\right) \]

       3. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{\color{blue}{2}}\right)}}\right) \]

   15. Applied rewrites 61.4%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}}\right)}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 76.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}\\ t_1 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot t\_0\\ t_2 := \mathsf{fma}\left(\cos \phi_2, t\_0, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\phi_2 \cdot 0.5\right)\right)\right)\\ t_3 := R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_2}}{\sqrt{1 - t\_2}}\right)\\ \mathbf{if}\;\phi_2 \leq -2.9 \cdot 10^{+48}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;\phi_2 \leq 0.00085:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_1}}{\sqrt{1 - t\_1}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (pow
          (fma
           (cos (* -0.5 lambda2))
           (sin (* 0.5 lambda1))
           (* (cos (* 0.5 lambda1)) (sin (* -0.5 lambda2))))
          2.0))
        (t_1 (+ (pow (sin (/ (- phi1 phi2) 2.0)) 2.0) (* (cos phi1) t_0)))
        (t_2 (fma (cos phi2) t_0 (- 0.5 (* 0.5 (cos (* 2.0 (* phi2 0.5)))))))
        (t_3 (* R (* 2.0 (atan2 (sqrt t_2) (sqrt (- 1.0 t_2)))))))
   (if (<= phi2 -2.9e+48)
     t_3
     (if (<= phi2 0.00085)
       (* R (* 2.0 (atan2 (sqrt t_1) (sqrt (- 1.0 t_1)))))
       t_3))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = pow(fma(cos((-0.5 * lambda2)), sin((0.5 * lambda1)), (cos((0.5 * lambda1)) * sin((-0.5 * lambda2)))), 2.0);
	double t_1 = pow(sin(((phi1 - phi2) / 2.0)), 2.0) + (cos(phi1) * t_0);
	double t_2 = fma(cos(phi2), t_0, (0.5 - (0.5 * cos((2.0 * (phi2 * 0.5))))));
	double t_3 = R * (2.0 * atan2(sqrt(t_2), sqrt((1.0 - t_2))));
	double tmp;
	if (phi2 <= -2.9e+48) {
		tmp = t_3;
	} else if (phi2 <= 0.00085) {
		tmp = R * (2.0 * atan2(sqrt(t_1), sqrt((1.0 - t_1))));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = fma(cos(Float64(-0.5 * lambda2)), sin(Float64(0.5 * lambda1)), Float64(cos(Float64(0.5 * lambda1)) * sin(Float64(-0.5 * lambda2)))) ^ 2.0
	t_1 = Float64((sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0) + Float64(cos(phi1) * t_0))
	t_2 = fma(cos(phi2), t_0, Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(phi2 * 0.5))))))
	t_3 = Float64(R * Float64(2.0 * atan(sqrt(t_2), sqrt(Float64(1.0 - t_2)))))
	tmp = 0.0
	if (phi2 <= -2.9e+48)
		tmp = t_3;
	elseif (phi2 <= 0.00085)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_1), sqrt(Float64(1.0 - t_1)))));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Power[N[(N[Cos[N[(-0.5 * lambda2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision] + N[(N[Cos[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(-0.5 * lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[phi2], $MachinePrecision] * t$95$0 + N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(phi2 * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$2], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$2), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -2.9e+48], t$95$3, If[LessEqual[phi2, 0.00085], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$1], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -2.8999999999999999e48 or 8.49999999999999953e-4 < phi2

   1. Initial program 62.0%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 61.1%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 62.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 62.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}}\right) \]

   9. Applied rewrites 77.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right)}}\right) \]
   10. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       7. mult-flip-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       8. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       9. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       10. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       11. lift-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       12. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       13. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       14. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       15. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       17. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       18. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       19. mult-flip-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       20. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       21. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       22. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       23. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       24. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       25. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   11. Applied rewrites 78.2%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   12. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       7. mult-flip-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       8. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       9. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       10. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       11. lift-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       12. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       13. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       14. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       15. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       17. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       18. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       19. mult-flip-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       20. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       21. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       22. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       23. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       24. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       25. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   13. Applied rewrites 98.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   14. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}}}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   15. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \color{blue}{{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   16. Applied rewrites 57.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   17. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}}\right) \]
   18. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, \color{blue}{{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

   19. Applied rewrites 56.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)}}}\right) \]
   20. Step-by-step derivation

       1. lift-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       2. unpow2 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       3. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       4. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       5. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \sin \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       6. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \sin \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       7. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \sin \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       8. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \sin \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       9. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \sin \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       10. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \sin \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       11. sqr-sin-a N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       12. lower--.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       13. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       14. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       15. lower-*.f64 55.3

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\phi_2 \cdot 0.5\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)}}\right) \]

   21. Applied rewrites 55.3%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\phi_2 \cdot 0.5\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)}}\right) \]
   22. Step-by-step derivation

       1. lift-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       2. unpow2 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}}\right) \]

       3. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}}\right) \]

       4. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}}\right) \]

       5. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \sin \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}}\right) \]

       6. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \sin \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}}\right) \]

       7. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \sin \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}}\right) \]

       8. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \sin \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}}\right) \]

       9. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \sin \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}}\right) \]

       10. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \sin \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}}\right) \]

       11. sqr-sin-a N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

       12. lower--.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

       13. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

       14. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

       15. lower-*.f64 55.3

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\phi_2 \cdot 0.5\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\phi_2 \cdot 0.5\right)\right)\right)}}\right) \]

   23. Applied rewrites 55.3%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\phi_2 \cdot 0.5\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\phi_2 \cdot 0.5\right)\right)\right)}}\right) \]

   if -2.8999999999999999e48 < phi2 < 8.49999999999999953e-4

   1. Initial program 62.0%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 61.1%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 62.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 62.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}}\right) \]

   9. Applied rewrites 77.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right)}}\right) \]

   10. Taylor expanded in phi2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   11. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot \color{blue}{{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\color{blue}{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{\color{blue}{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   12. Applied rewrites 63.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   13. Taylor expanded in phi2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}\right)}}\right) \]
   14. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot \color{blue}{{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}\right)}}\right) \]

       2. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\color{blue}{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}}^{2}\right)}}\right) \]

       3. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{\color{blue}{2}}\right)}}\right) \]

   15. Applied rewrites 61.4%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}}\right)}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 76.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := \sin \left(-0.5 \cdot \lambda_2\right)\\ t_1 := \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(t\_0, \cos \left(-0.5 \cdot \lambda_1\right), \sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)\\ t_2 := \mathsf{fma}\left(\cos \phi_1, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot t\_0\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)\\ t_3 := R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_2}}{\sqrt{1 - t\_2}}\right)\\ \mathbf{if}\;\phi_1 \leq -3.3 \cdot 10^{-10}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;\phi_1 \leq 6000000000:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_1}}{\sqrt{1 - t\_1}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (* -0.5 lambda2)))
        (t_1
         (fma
          (cos phi2)
          (pow
           (fma
            t_0
            (cos (* -0.5 lambda1))
            (* (sin (* lambda1 0.5)) (cos (* lambda2 0.5))))
           2.0)
          (pow (sin (* 0.5 phi2)) 2.0)))
        (t_2
         (fma
          (cos phi1)
          (pow
           (fma
            (cos (* -0.5 lambda2))
            (sin (* 0.5 lambda1))
            (* (cos (* 0.5 lambda1)) t_0))
           2.0)
          (pow (sin (* 0.5 phi1)) 2.0)))
        (t_3 (* R (* 2.0 (atan2 (sqrt t_2) (sqrt (- 1.0 t_2)))))))
   (if (<= phi1 -3.3e-10)
     t_3
     (if (<= phi1 6000000000.0)
       (* R (* 2.0 (atan2 (sqrt t_1) (sqrt (- 1.0 t_1)))))
       t_3))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin((-0.5 * lambda2));
	double t_1 = fma(cos(phi2), pow(fma(t_0, cos((-0.5 * lambda1)), (sin((lambda1 * 0.5)) * cos((lambda2 * 0.5)))), 2.0), pow(sin((0.5 * phi2)), 2.0));
	double t_2 = fma(cos(phi1), pow(fma(cos((-0.5 * lambda2)), sin((0.5 * lambda1)), (cos((0.5 * lambda1)) * t_0)), 2.0), pow(sin((0.5 * phi1)), 2.0));
	double t_3 = R * (2.0 * atan2(sqrt(t_2), sqrt((1.0 - t_2))));
	double tmp;
	if (phi1 <= -3.3e-10) {
		tmp = t_3;
	} else if (phi1 <= 6000000000.0) {
		tmp = R * (2.0 * atan2(sqrt(t_1), sqrt((1.0 - t_1))));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(-0.5 * lambda2))
	t_1 = fma(cos(phi2), (fma(t_0, cos(Float64(-0.5 * lambda1)), Float64(sin(Float64(lambda1 * 0.5)) * cos(Float64(lambda2 * 0.5)))) ^ 2.0), (sin(Float64(0.5 * phi2)) ^ 2.0))
	t_2 = fma(cos(phi1), (fma(cos(Float64(-0.5 * lambda2)), sin(Float64(0.5 * lambda1)), Float64(cos(Float64(0.5 * lambda1)) * t_0)) ^ 2.0), (sin(Float64(0.5 * phi1)) ^ 2.0))
	t_3 = Float64(R * Float64(2.0 * atan(sqrt(t_2), sqrt(Float64(1.0 - t_2)))))
	tmp = 0.0
	if (phi1 <= -3.3e-10)
		tmp = t_3;
	elseif (phi1 <= 6000000000.0)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_1), sqrt(Float64(1.0 - t_1)))));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(-0.5 * lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[phi2], $MachinePrecision] * N[Power[N[(t$95$0 * N[Cos[N[(-0.5 * lambda1), $MachinePrecision]], $MachinePrecision] + N[(N[Sin[N[(lambda1 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(lambda2 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[phi1], $MachinePrecision] * N[Power[N[(N[Cos[N[(-0.5 * lambda2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision] + N[(N[Cos[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$2], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$2), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi1, -3.3e-10], t$95$3, If[LessEqual[phi1, 6000000000.0], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$1], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -3.3e-10 or 6e9 < phi1

   1. Initial program 62.0%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 61.1%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 62.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 62.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}}\right) \]

   9. Applied rewrites 77.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right)}}\right) \]

   10. Taylor expanded in phi2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   11. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   12. Applied rewrites 57.0%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   13. Taylor expanded in phi2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   14. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   15. Applied rewrites 57.1%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]

   if -3.3e-10 < phi1 < 6e9

   1. Initial program 62.0%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 61.1%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 62.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 62.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}}\right) \]

   9. Applied rewrites 77.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right)}}\right) \]
   10. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       7. mult-flip-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       8. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       9. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       10. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       11. lift-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       12. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       13. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       14. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       15. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       17. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       18. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       19. mult-flip-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       20. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       21. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       22. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       23. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       24. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       25. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   11. Applied rewrites 78.2%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   12. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       7. mult-flip-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       8. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       9. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       10. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       11. lift-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       12. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       13. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       14. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       15. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       17. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       18. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       19. mult-flip-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       20. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       21. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       22. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       23. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       24. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       25. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   13. Applied rewrites 98.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   14. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}}}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   15. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \color{blue}{{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   16. Applied rewrites 57.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   17. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}}\right) \]
   18. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, \color{blue}{{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

   19. Applied rewrites 56.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)}}}\right) \]
   20. Step-by-step derivation

       1. lift-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       2. +-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right) + \cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       3. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right) + \cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       4. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\sin \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \cos \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       5. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\sin \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \cos \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       6. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\sin \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \cos \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       7. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\sin \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \cos \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\lambda_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       8. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\sin \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \cos \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\lambda_2 \cdot \frac{1}{-2}\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       9. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\sin \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \cos \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{\lambda_2}{-2}\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       10. lift-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\sin \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \cos \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{\lambda_2}{-2}\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       11. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\sin \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \cos \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{\lambda_2}{-2}\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       12. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\sin \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \cos \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{\lambda_2}{-2}\right) \cdot \sin \left(\lambda_1 \cdot \frac{1}{2}\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       13. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\sin \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \cos \left(\frac{1}{2} \cdot \lambda_1\right) + \sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{-2}\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       14. lower-fma.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \lambda_2\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right), \sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{-2}\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

   21. Applied rewrites 56.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\sin \left(-0.5 \cdot \lambda_2\right), \cos \left(-0.5 \cdot \lambda_1\right), \sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)}}\right) \]
   22. Step-by-step derivation

       1. lift-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \lambda_2\right), \cos \left(\frac{-1}{2} \cdot \lambda_1\right), \sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       2. +-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \lambda_2\right), \cos \left(\frac{-1}{2} \cdot \lambda_1\right), \sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right) + \cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       3. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \lambda_2\right), \cos \left(\frac{-1}{2} \cdot \lambda_1\right), \sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right) + \cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       4. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \lambda_2\right), \cos \left(\frac{-1}{2} \cdot \lambda_1\right), \sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\sin \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \cos \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       5. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \lambda_2\right), \cos \left(\frac{-1}{2} \cdot \lambda_1\right), \sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\sin \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \cos \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       6. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \lambda_2\right), \cos \left(\frac{-1}{2} \cdot \lambda_1\right), \sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\sin \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \cos \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       7. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \lambda_2\right), \cos \left(\frac{-1}{2} \cdot \lambda_1\right), \sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\sin \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \cos \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\lambda_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       8. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \lambda_2\right), \cos \left(\frac{-1}{2} \cdot \lambda_1\right), \sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\sin \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \cos \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\lambda_2 \cdot \frac{1}{-2}\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       9. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \lambda_2\right), \cos \left(\frac{-1}{2} \cdot \lambda_1\right), \sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\sin \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \cos \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{\lambda_2}{-2}\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       10. lift-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \lambda_2\right), \cos \left(\frac{-1}{2} \cdot \lambda_1\right), \sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\sin \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \cos \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{\lambda_2}{-2}\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       11. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \lambda_2\right), \cos \left(\frac{-1}{2} \cdot \lambda_1\right), \sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\sin \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \cos \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{\lambda_2}{-2}\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       12. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \lambda_2\right), \cos \left(\frac{-1}{2} \cdot \lambda_1\right), \sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\sin \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \cos \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{\lambda_2}{-2}\right) \cdot \sin \left(\lambda_1 \cdot \frac{1}{2}\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       13. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \lambda_2\right), \cos \left(\frac{-1}{2} \cdot \lambda_1\right), \sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\sin \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \cos \left(\frac{1}{2} \cdot \lambda_1\right) + \sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{-2}\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       14. lower-fma.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \lambda_2\right), \cos \left(\frac{-1}{2} \cdot \lambda_1\right), \sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \lambda_2\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right), \sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{-2}\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

   23. Applied rewrites 56.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\sin \left(-0.5 \cdot \lambda_2\right), \cos \left(-0.5 \cdot \lambda_1\right), \sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\sin \left(-0.5 \cdot \lambda_2\right), \cos \left(-0.5 \cdot \lambda_1\right), \sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 76.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}\\ t_1 := \mathsf{fma}\left(\cos \phi_2, t\_0, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)\\ t_2 := \mathsf{fma}\left(\cos \phi_1, t\_0, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)\\ t_3 := R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_2}}{\sqrt{1 - t\_2}}\right)\\ \mathbf{if}\;\phi_1 \leq -3.3 \cdot 10^{-10}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;\phi_1 \leq 6000000000:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_1}}{\sqrt{1 - t\_1}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (pow
          (fma
           (cos (* -0.5 lambda2))
           (sin (* 0.5 lambda1))
           (* (cos (* 0.5 lambda1)) (sin (* -0.5 lambda2))))
          2.0))
        (t_1 (fma (cos phi2) t_0 (pow (sin (* -0.5 phi2)) 2.0)))
        (t_2 (fma (cos phi1) t_0 (pow (sin (* 0.5 phi1)) 2.0)))
        (t_3 (* R (* 2.0 (atan2 (sqrt t_2) (sqrt (- 1.0 t_2)))))))
   (if (<= phi1 -3.3e-10)
     t_3
     (if (<= phi1 6000000000.0)
       (* R (* 2.0 (atan2 (sqrt t_1) (sqrt (- 1.0 t_1)))))
       t_3))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = pow(fma(cos((-0.5 * lambda2)), sin((0.5 * lambda1)), (cos((0.5 * lambda1)) * sin((-0.5 * lambda2)))), 2.0);
	double t_1 = fma(cos(phi2), t_0, pow(sin((-0.5 * phi2)), 2.0));
	double t_2 = fma(cos(phi1), t_0, pow(sin((0.5 * phi1)), 2.0));
	double t_3 = R * (2.0 * atan2(sqrt(t_2), sqrt((1.0 - t_2))));
	double tmp;
	if (phi1 <= -3.3e-10) {
		tmp = t_3;
	} else if (phi1 <= 6000000000.0) {
		tmp = R * (2.0 * atan2(sqrt(t_1), sqrt((1.0 - t_1))));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = fma(cos(Float64(-0.5 * lambda2)), sin(Float64(0.5 * lambda1)), Float64(cos(Float64(0.5 * lambda1)) * sin(Float64(-0.5 * lambda2)))) ^ 2.0
	t_1 = fma(cos(phi2), t_0, (sin(Float64(-0.5 * phi2)) ^ 2.0))
	t_2 = fma(cos(phi1), t_0, (sin(Float64(0.5 * phi1)) ^ 2.0))
	t_3 = Float64(R * Float64(2.0 * atan(sqrt(t_2), sqrt(Float64(1.0 - t_2)))))
	tmp = 0.0
	if (phi1 <= -3.3e-10)
		tmp = t_3;
	elseif (phi1 <= 6000000000.0)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_1), sqrt(Float64(1.0 - t_1)))));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Power[N[(N[Cos[N[(-0.5 * lambda2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision] + N[(N[Cos[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(-0.5 * lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[phi2], $MachinePrecision] * t$95$0 + N[Power[N[Sin[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[phi1], $MachinePrecision] * t$95$0 + N[Power[N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$2], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$2), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi1, -3.3e-10], t$95$3, If[LessEqual[phi1, 6000000000.0], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$1], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -3.3e-10 or 6e9 < phi1

   1. Initial program 62.0%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 61.1%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 62.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 62.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}}\right) \]

   9. Applied rewrites 77.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right)}}\right) \]

   10. Taylor expanded in phi2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   11. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   12. Applied rewrites 57.0%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   13. Taylor expanded in phi2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   14. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   15. Applied rewrites 57.1%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]

   if -3.3e-10 < phi1 < 6e9

   1. Initial program 62.0%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 61.1%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 62.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 62.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}}\right) \]

   9. Applied rewrites 77.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right)}}\right) \]

   10. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   11. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \color{blue}{{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   12. Applied rewrites 56.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   13. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}}\right) \]
   14. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, \color{blue}{{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

   15. Applied rewrites 56.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 76.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}\\ t_1 := \mathsf{fma}\left(\cos \phi_1, t\_0, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)\\ t_2 := \mathsf{fma}\left(\cos \phi_2, t\_0, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\phi_2 \cdot 0.5\right)\right)\right)\\ t_3 := R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_2}}{\sqrt{1 - t\_2}}\right)\\ \mathbf{if}\;\phi_2 \leq -1.15 \cdot 10^{-5}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;\phi_2 \leq 5 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_1}}{\sqrt{1 - t\_1}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (pow
          (fma
           (cos (* -0.5 lambda2))
           (sin (* 0.5 lambda1))
           (* (cos (* 0.5 lambda1)) (sin (* -0.5 lambda2))))
          2.0))
        (t_1 (fma (cos phi1) t_0 (pow (sin (* 0.5 phi1)) 2.0)))
        (t_2 (fma (cos phi2) t_0 (- 0.5 (* 0.5 (cos (* 2.0 (* phi2 0.5)))))))
        (t_3 (* R (* 2.0 (atan2 (sqrt t_2) (sqrt (- 1.0 t_2)))))))
   (if (<= phi2 -1.15e-5)
     t_3
     (if (<= phi2 5e-5)
       (* R (* 2.0 (atan2 (sqrt t_1) (sqrt (- 1.0 t_1)))))
       t_3))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = pow(fma(cos((-0.5 * lambda2)), sin((0.5 * lambda1)), (cos((0.5 * lambda1)) * sin((-0.5 * lambda2)))), 2.0);
	double t_1 = fma(cos(phi1), t_0, pow(sin((0.5 * phi1)), 2.0));
	double t_2 = fma(cos(phi2), t_0, (0.5 - (0.5 * cos((2.0 * (phi2 * 0.5))))));
	double t_3 = R * (2.0 * atan2(sqrt(t_2), sqrt((1.0 - t_2))));
	double tmp;
	if (phi2 <= -1.15e-5) {
		tmp = t_3;
	} else if (phi2 <= 5e-5) {
		tmp = R * (2.0 * atan2(sqrt(t_1), sqrt((1.0 - t_1))));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = fma(cos(Float64(-0.5 * lambda2)), sin(Float64(0.5 * lambda1)), Float64(cos(Float64(0.5 * lambda1)) * sin(Float64(-0.5 * lambda2)))) ^ 2.0
	t_1 = fma(cos(phi1), t_0, (sin(Float64(0.5 * phi1)) ^ 2.0))
	t_2 = fma(cos(phi2), t_0, Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(phi2 * 0.5))))))
	t_3 = Float64(R * Float64(2.0 * atan(sqrt(t_2), sqrt(Float64(1.0 - t_2)))))
	tmp = 0.0
	if (phi2 <= -1.15e-5)
		tmp = t_3;
	elseif (phi2 <= 5e-5)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_1), sqrt(Float64(1.0 - t_1)))));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Power[N[(N[Cos[N[(-0.5 * lambda2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision] + N[(N[Cos[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(-0.5 * lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[phi1], $MachinePrecision] * t$95$0 + N[Power[N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[phi2], $MachinePrecision] * t$95$0 + N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(phi2 * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$2], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$2), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -1.15e-5], t$95$3, If[LessEqual[phi2, 5e-5], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$1], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -1.15e-5 or 5.00000000000000024e-5 < phi2

   1. Initial program 62.0%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 61.1%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 62.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 62.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}}\right) \]

   9. Applied rewrites 77.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right)}}\right) \]
   10. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       7. mult-flip-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       8. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       9. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       10. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       11. lift-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       12. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       13. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       14. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       15. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       17. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       18. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       19. mult-flip-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       20. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       21. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       22. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       23. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       24. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       25. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   11. Applied rewrites 78.2%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   12. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       7. mult-flip-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       8. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       9. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       10. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       11. lift-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       12. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       13. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       14. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       15. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       17. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       18. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       19. mult-flip-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       20. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       21. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       22. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       23. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       24. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       25. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   13. Applied rewrites 98.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   14. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}}}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   15. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \color{blue}{{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   16. Applied rewrites 57.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   17. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}}\right) \]
   18. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, \color{blue}{{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

   19. Applied rewrites 56.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)}}}\right) \]
   20. Step-by-step derivation

       1. lift-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       2. unpow2 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       3. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       4. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       5. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \sin \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       6. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \sin \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       7. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \sin \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       8. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \sin \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       9. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \sin \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       10. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \sin \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       11. sqr-sin-a N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       12. lower--.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       13. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       14. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       15. lower-*.f64 55.3

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\phi_2 \cdot 0.5\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)}}\right) \]

   21. Applied rewrites 55.3%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\phi_2 \cdot 0.5\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)}}\right) \]
   22. Step-by-step derivation

       1. lift-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       2. unpow2 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}}\right) \]

       3. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}}\right) \]

       4. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}}\right) \]

       5. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \sin \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}}\right) \]

       6. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \sin \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}}\right) \]

       7. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \sin \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}}\right) \]

       8. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \sin \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}}\right) \]

       9. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \sin \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}}\right) \]

       10. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \sin \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}}\right) \]

       11. sqr-sin-a N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

       12. lower--.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

       13. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

       14. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

       15. lower-*.f64 55.3

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\phi_2 \cdot 0.5\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\phi_2 \cdot 0.5\right)\right)\right)}}\right) \]

   23. Applied rewrites 55.3%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\phi_2 \cdot 0.5\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\phi_2 \cdot 0.5\right)\right)\right)}}\right) \]

   if -1.15e-5 < phi2 < 5.00000000000000024e-5

   1. Initial program 62.0%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 61.1%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 62.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 62.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}}\right) \]

   9. Applied rewrites 77.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right)}}\right) \]

   10. Taylor expanded in phi2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   11. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   12. Applied rewrites 57.0%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   13. Taylor expanded in phi2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   14. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   15. Applied rewrites 57.1%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 68.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\\ t_1 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_2 := \cos \left(\left(\phi_2 - \phi_1\right) \cdot 0.5\right)\\ t_3 := \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\phi_2 \cdot 0.5\right)\right)\right)\\ t_4 := 0.5 \cdot \left(\lambda_1 - \lambda_2\right)\\ t_5 := t\_0 + \cos \phi_1 \cdot {\sin t\_4}^{2}\\ \mathbf{if}\;\phi_1 \leq -3.3 \cdot 10^{-10}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_5}}{\sqrt{1 - t\_5}}\right)\\ \mathbf{elif}\;\phi_1 \leq 9.5 \cdot 10^{-96}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_3}}{\sqrt{1 - t\_3}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_0 + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_1\right) \cdot t\_1}}{\sqrt{\mathsf{fma}\left(t\_2, t\_2, \left(-\cos \phi_2 \cdot \cos \phi_1\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot t\_4\right)\right)\right)}}\right)\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (pow (sin (/ (- phi1 phi2) 2.0)) 2.0))
        (t_1 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_2 (cos (* (- phi2 phi1) 0.5)))
        (t_3
         (fma
          (cos phi2)
          (pow
           (fma
            (cos (* -0.5 lambda2))
            (sin (* 0.5 lambda1))
            (* (cos (* 0.5 lambda1)) (sin (* -0.5 lambda2))))
           2.0)
          (- 0.5 (* 0.5 (cos (* 2.0 (* phi2 0.5)))))))
        (t_4 (* 0.5 (- lambda1 lambda2)))
        (t_5 (+ t_0 (* (cos phi1) (pow (sin t_4) 2.0)))))
   (if (<= phi1 -3.3e-10)
     (* R (* 2.0 (atan2 (sqrt t_5) (sqrt (- 1.0 t_5)))))
     (if (<= phi1 9.5e-96)
       (* R (* 2.0 (atan2 (sqrt t_3) (sqrt (- 1.0 t_3)))))
       (*
        R
        (*
         2.0
         (atan2
          (sqrt (+ t_0 (* (* (* (cos phi1) (cos phi2)) t_1) t_1)))
          (sqrt
           (fma
            t_2
            t_2
            (*
             (- (* (cos phi2) (cos phi1)))
             (- 0.5 (* 0.5 (cos (* 2.0 t_4))))))))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = pow(sin(((phi1 - phi2) / 2.0)), 2.0);
	double t_1 = sin(((lambda1 - lambda2) / 2.0));
	double t_2 = cos(((phi2 - phi1) * 0.5));
	double t_3 = fma(cos(phi2), pow(fma(cos((-0.5 * lambda2)), sin((0.5 * lambda1)), (cos((0.5 * lambda1)) * sin((-0.5 * lambda2)))), 2.0), (0.5 - (0.5 * cos((2.0 * (phi2 * 0.5))))));
	double t_4 = 0.5 * (lambda1 - lambda2);
	double t_5 = t_0 + (cos(phi1) * pow(sin(t_4), 2.0));
	double tmp;
	if (phi1 <= -3.3e-10) {
		tmp = R * (2.0 * atan2(sqrt(t_5), sqrt((1.0 - t_5))));
	} else if (phi1 <= 9.5e-96) {
		tmp = R * (2.0 * atan2(sqrt(t_3), sqrt((1.0 - t_3))));
	} else {
		tmp = R * (2.0 * atan2(sqrt((t_0 + (((cos(phi1) * cos(phi2)) * t_1) * t_1))), sqrt(fma(t_2, t_2, (-(cos(phi2) * cos(phi1)) * (0.5 - (0.5 * cos((2.0 * t_4)))))))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0
	t_1 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_2 = cos(Float64(Float64(phi2 - phi1) * 0.5))
	t_3 = fma(cos(phi2), (fma(cos(Float64(-0.5 * lambda2)), sin(Float64(0.5 * lambda1)), Float64(cos(Float64(0.5 * lambda1)) * sin(Float64(-0.5 * lambda2)))) ^ 2.0), Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(phi2 * 0.5))))))
	t_4 = Float64(0.5 * Float64(lambda1 - lambda2))
	t_5 = Float64(t_0 + Float64(cos(phi1) * (sin(t_4) ^ 2.0)))
	tmp = 0.0
	if (phi1 <= -3.3e-10)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_5), sqrt(Float64(1.0 - t_5)))));
	elseif (phi1 <= 9.5e-96)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_3), sqrt(Float64(1.0 - t_3)))));
	else
		tmp = Float64(R * Float64(2.0 * atan(sqrt(Float64(t_0 + Float64(Float64(Float64(cos(phi1) * cos(phi2)) * t_1) * t_1))), sqrt(fma(t_2, t_2, Float64(Float64(-Float64(cos(phi2) * cos(phi1))) * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * t_4))))))))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(N[(phi2 - phi1), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[Cos[phi2], $MachinePrecision] * N[Power[N[(N[Cos[N[(-0.5 * lambda2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision] + N[(N[Cos[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(-0.5 * lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(phi2 * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$0 + N[(N[Cos[phi1], $MachinePrecision] * N[Power[N[Sin[t$95$4], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi1, -3.3e-10], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$5], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi1, 9.5e-96], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$3], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$3), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(t$95$0 + N[(N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(t$95$2 * t$95$2 + N[((-N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]) * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * t$95$4), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -3.3e-10

   1. Initial program 62.0%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 53.2

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 53.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}\right)}}\right) \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}\right) \]

      6. lower--.f64 51.1

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 51.1%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}\right)}}\right) \]

   if -3.3e-10 < phi1 < 9.4999999999999993e-96

   1. Initial program 62.0%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 61.1%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 62.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 62.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}}\right) \]

   9. Applied rewrites 77.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right)}}\right) \]
   10. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       7. mult-flip-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       8. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       9. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       10. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       11. lift-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       12. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       13. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       14. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       15. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       17. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       18. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       19. mult-flip-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       20. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       21. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       22. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       23. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       24. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       25. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   11. Applied rewrites 78.2%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   12. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       7. mult-flip-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       8. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       9. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       10. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       11. lift-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       12. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       13. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       14. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       15. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       17. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       18. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       19. mult-flip-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       20. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       21. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       22. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       23. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       24. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       25. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   13. Applied rewrites 98.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   14. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}}}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   15. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \color{blue}{{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   16. Applied rewrites 57.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   17. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}}\right) \]
   18. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, \color{blue}{{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

   19. Applied rewrites 56.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)}}}\right) \]
   20. Step-by-step derivation

       1. lift-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       2. unpow2 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       3. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       4. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       5. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \sin \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       6. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \sin \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       7. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \sin \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       8. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \sin \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       9. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \sin \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       10. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \sin \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       11. sqr-sin-a N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       12. lower--.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       13. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       14. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       15. lower-*.f64 55.3

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\phi_2 \cdot 0.5\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)}}\right) \]

   21. Applied rewrites 55.3%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\phi_2 \cdot 0.5\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)}}\right) \]
   22. Step-by-step derivation

       1. lift-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       2. unpow2 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}}\right) \]

       3. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}}\right) \]

       4. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}}\right) \]

       5. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \sin \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}}\right) \]

       6. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \sin \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}}\right) \]

       7. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \sin \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}}\right) \]

       8. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \sin \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}}\right) \]

       9. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \sin \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}}\right) \]

       10. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \sin \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}}\right) \]

       11. sqr-sin-a N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

       12. lower--.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

       13. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

       14. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

       15. lower-*.f64 55.3

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\phi_2 \cdot 0.5\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\phi_2 \cdot 0.5\right)\right)\right)}}\right) \]

   23. Applied rewrites 55.3%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\phi_2 \cdot 0.5\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\phi_2 \cdot 0.5\right)\right)\right)}}\right) \]

   if 9.4999999999999993e-96 < phi1

   1. Initial program 62.0%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 62.1%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \left(\left(\phi_2 - \phi_1\right) \cdot 0.5\right), \cos \left(\left(\phi_2 - \phi_1\right) \cdot 0.5\right), \left(-\cos \phi_2 \cdot \cos \phi_1\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\right)}}}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 18: 62.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_0 := 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\phi_2 \cdot 0.5\right)\right)\\ t_1 := 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\\ t_2 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\\ t_3 := \mathsf{fma}\left(t\_1, \cos \phi_2, t\_0\right)\\ t_4 := t\_0 + t\_1 \cdot \cos \phi_2\\ \mathbf{if}\;\phi_2 \leq -9.5 \cdot 10^{+47}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_4}}{\sqrt{1 - t\_4}}\right)\\ \mathbf{elif}\;\phi_2 \leq 0.00085:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_2}}{\sqrt{1 - t\_2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\tan^{-1}_* \frac{\sqrt{t\_3}}{\sqrt{1 - t\_3}} \cdot 2\right) \cdot R\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (- 0.5 (* 0.5 (cos (* 2.0 (* phi2 0.5))))))
        (t_1 (- 0.5 (* 0.5 (cos (* 2.0 (* (- lambda1 lambda2) 0.5))))))
        (t_2
         (+
          (pow (sin (/ (- phi1 phi2) 2.0)) 2.0)
          (* (cos phi1) (pow (sin (* 0.5 (- lambda1 lambda2))) 2.0))))
        (t_3 (fma t_1 (cos phi2) t_0))
        (t_4 (+ t_0 (* t_1 (cos phi2)))))
   (if (<= phi2 -9.5e+47)
     (* R (* 2.0 (atan2 (sqrt t_4) (sqrt (- 1.0 t_4)))))
     (if (<= phi2 0.00085)
       (* R (* 2.0 (atan2 (sqrt t_2) (sqrt (- 1.0 t_2)))))
       (* (* (atan2 (sqrt t_3) (sqrt (- 1.0 t_3))) 2.0) R)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = 0.5 - (0.5 * cos((2.0 * (phi2 * 0.5))));
	double t_1 = 0.5 - (0.5 * cos((2.0 * ((lambda1 - lambda2) * 0.5))));
	double t_2 = pow(sin(((phi1 - phi2) / 2.0)), 2.0) + (cos(phi1) * pow(sin((0.5 * (lambda1 - lambda2))), 2.0));
	double t_3 = fma(t_1, cos(phi2), t_0);
	double t_4 = t_0 + (t_1 * cos(phi2));
	double tmp;
	if (phi2 <= -9.5e+47) {
		tmp = R * (2.0 * atan2(sqrt(t_4), sqrt((1.0 - t_4))));
	} else if (phi2 <= 0.00085) {
		tmp = R * (2.0 * atan2(sqrt(t_2), sqrt((1.0 - t_2))));
	} else {
		tmp = (atan2(sqrt(t_3), sqrt((1.0 - t_3))) * 2.0) * R;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(phi2 * 0.5)))))
	t_1 = Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(Float64(lambda1 - lambda2) * 0.5)))))
	t_2 = Float64((sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0) + Float64(cos(phi1) * (sin(Float64(0.5 * Float64(lambda1 - lambda2))) ^ 2.0)))
	t_3 = fma(t_1, cos(phi2), t_0)
	t_4 = Float64(t_0 + Float64(t_1 * cos(phi2)))
	tmp = 0.0
	if (phi2 <= -9.5e+47)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_4), sqrt(Float64(1.0 - t_4)))));
	elseif (phi2 <= 0.00085)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_2), sqrt(Float64(1.0 - t_2)))));
	else
		tmp = Float64(Float64(atan(sqrt(t_3), sqrt(Float64(1.0 - t_3))) * 2.0) * R);
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(phi2 * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(N[(lambda1 - lambda2), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * N[Power[N[Sin[N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 * N[Cos[phi2], $MachinePrecision] + t$95$0), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$0 + N[(t$95$1 * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -9.5e+47], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$4], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$4), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 0.00085], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$2], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$2), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[ArcTan[N[Sqrt[t$95$3], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$3), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * R), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -9.50000000000000001e47

   1. Initial program 62.0%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 61.1%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 62.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 62.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}}\right) \]

   9. Applied rewrites 77.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right)}}\right) \]
   10. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       7. mult-flip-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       8. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       9. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       10. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       11. lift-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       12. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       13. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       14. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       15. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       17. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       18. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       19. mult-flip-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       20. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       21. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       22. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       23. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       24. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       25. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   11. Applied rewrites 78.2%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   12. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       7. mult-flip-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       8. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       9. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       10. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       11. lift-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       12. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       13. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       14. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       15. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       17. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       18. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       19. mult-flip-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       20. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       21. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       22. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       23. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       24. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       25. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   13. Applied rewrites 98.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   14. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}}}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   15. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \color{blue}{{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   16. Applied rewrites 57.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   17. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}}\right) \]
   18. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, \color{blue}{{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

   19. Applied rewrites 56.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)}}}\right) \]

   21. Applied rewrites 42.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\phi_2 \cdot 0.5\right)\right)\right) + \color{blue}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right) \cdot \cos \phi_2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)}}\right) \]

   23. Applied rewrites 42.4%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\phi_2 \cdot 0.5\right)\right)\right) + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right) \cdot \cos \phi_2}}{\sqrt{1 - \left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\phi_2 \cdot 0.5\right)\right)\right) + \color{blue}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right) \cdot \cos \phi_2}\right)}}\right) \]

   if -9.50000000000000001e47 < phi2 < 8.49999999999999953e-4

   1. Initial program 62.0%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 53.2

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 53.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}\right)}}\right) \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}\right) \]

      6. lower--.f64 51.1

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 51.1%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}\right)}}\right) \]

   if 8.49999999999999953e-4 < phi2

   1. Initial program 62.0%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 61.1%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 62.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 62.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}}\right) \]

   9. Applied rewrites 77.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right)}}\right) \]
   10. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       7. mult-flip-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       8. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       9. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       10. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       11. lift-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       12. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       13. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       14. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       15. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       17. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       18. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       19. mult-flip-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       20. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       21. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       22. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       23. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       24. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       25. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   11. Applied rewrites 78.2%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   12. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       7. mult-flip-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       8. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       9. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       10. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       11. lift-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       12. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       13. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       14. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       15. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       17. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       18. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       19. mult-flip-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       20. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       21. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       22. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       23. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       24. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       25. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   13. Applied rewrites 98.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   14. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}}}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   15. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \color{blue}{{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   16. Applied rewrites 57.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   17. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}}\right) \]
   18. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, \color{blue}{{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

   19. Applied rewrites 56.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)}}}\right) \]

   21. Applied rewrites 42.4%

       \[\leadsto \color{blue}{\left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right), \cos \phi_2, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\phi_2 \cdot 0.5\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right), \cos \phi_2, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\phi_2 \cdot 0.5\right)\right)\right)}} \cdot 2\right) \cdot R} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 19: 62.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} t_0 := {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\\ t_1 := 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\\ t_2 := R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, t\_0, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(t\_1, \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}\right)\\ \mathbf{if}\;\phi_1 \leq -0.42:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\phi_1 \leq 7.3:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot t\_0}}{\sqrt{\left(1 - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)\right) - t\_1 \cdot \cos \phi_2}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (pow (sin (* 0.5 (- lambda1 lambda2))) 2.0))
        (t_1 (- 0.5 (* 0.5 (cos (* 2.0 (* (- lambda1 lambda2) 0.5))))))
        (t_2
         (*
          R
          (*
           2.0
           (atan2
            (sqrt (fma (cos phi1) t_0 (pow (sin (* 0.5 phi1)) 2.0)))
            (sqrt
             (-
              1.0
              (fma
               t_1
               (cos phi1)
               (- 0.5 (* 0.5 (cos (* 2.0 (* 0.5 phi1)))))))))))))
   (if (<= phi1 -0.42)
     t_2
     (if (<= phi1 7.3)
       (*
        R
        (*
         2.0
         (atan2
          (sqrt (+ (pow (sin (/ (- phi1 phi2) 2.0)) 2.0) (* (cos phi2) t_0)))
          (sqrt
           (-
            (- 1.0 (- 0.5 (* 0.5 (cos (* 2.0 (* (- phi1 phi2) 0.5))))))
            (* t_1 (cos phi2)))))))
       t_2))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = pow(sin((0.5 * (lambda1 - lambda2))), 2.0);
	double t_1 = 0.5 - (0.5 * cos((2.0 * ((lambda1 - lambda2) * 0.5))));
	double t_2 = R * (2.0 * atan2(sqrt(fma(cos(phi1), t_0, pow(sin((0.5 * phi1)), 2.0))), sqrt((1.0 - fma(t_1, cos(phi1), (0.5 - (0.5 * cos((2.0 * (0.5 * phi1))))))))));
	double tmp;
	if (phi1 <= -0.42) {
		tmp = t_2;
	} else if (phi1 <= 7.3) {
		tmp = R * (2.0 * atan2(sqrt((pow(sin(((phi1 - phi2) / 2.0)), 2.0) + (cos(phi2) * t_0))), sqrt(((1.0 - (0.5 - (0.5 * cos((2.0 * ((phi1 - phi2) * 0.5)))))) - (t_1 * cos(phi2))))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(0.5 * Float64(lambda1 - lambda2))) ^ 2.0
	t_1 = Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(Float64(lambda1 - lambda2) * 0.5)))))
	t_2 = Float64(R * Float64(2.0 * atan(sqrt(fma(cos(phi1), t_0, (sin(Float64(0.5 * phi1)) ^ 2.0))), sqrt(Float64(1.0 - fma(t_1, cos(phi1), Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(0.5 * phi1)))))))))))
	tmp = 0.0
	if (phi1 <= -0.42)
		tmp = t_2;
	elseif (phi1 <= 7.3)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(Float64((sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0) + Float64(cos(phi2) * t_0))), sqrt(Float64(Float64(1.0 - Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(Float64(phi1 - phi2) * 0.5)))))) - Float64(t_1 * cos(phi2)))))));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Power[N[Sin[N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(N[(lambda1 - lambda2), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[Cos[phi1], $MachinePrecision] * t$95$0 + N[Power[N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(t$95$1 * N[Cos[phi1], $MachinePrecision] + N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(0.5 * phi1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi1, -0.42], t$95$2, If[LessEqual[phi1, 7.3], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[(1.0 - N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(N[(phi1 - phi2), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(t$95$1 * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -0.419999999999999984 or 7.29999999999999982 < phi1

   1. Initial program 62.0%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-*.f64 46.7

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 46.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   6. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      9. lower-*.f64 46.8

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 46.8%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]

   9. Applied rewrites 46.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\color{blue}{\sqrt{1 - \mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}}\right) \]

   if -0.419999999999999984 < phi1 < 7.29999999999999982

   1. Initial program 62.0%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 52.8

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 52.8%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_2 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}\right)}}\right) \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}\right) \]

      6. lower--.f64 50.7

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 50.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_2 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}\right)}}\right) \]

   9. Applied rewrites 50.8%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\color{blue}{\sqrt{\left(1 - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)\right) - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right) \cdot \cos \phi_2}}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 62.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := 0.5 \cdot \left(\lambda_1 - \lambda_2\right)\\ t_2 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\\ t_3 := t\_2 + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_0\right) \cdot t\_0\\ t_4 := \left|\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot t\_1\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)\right|\\ \mathbf{if}\;2 \cdot \tan^{-1}_* \frac{\sqrt{t\_3}}{\sqrt{1 - t\_3}} \leq 0.2:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_2 + \cos \phi_2 \cdot {\sin t\_1}^{2}}}{\sqrt{\left(1 - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)\right) - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right) \cdot \cos \phi_2}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_4}}{\sqrt{1 - t\_4}}\right)\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_1 (* 0.5 (- lambda1 lambda2)))
        (t_2 (pow (sin (/ (- phi1 phi2) 2.0)) 2.0))
        (t_3 (+ t_2 (* (* (* (cos phi1) (cos phi2)) t_0) t_0)))
        (t_4
         (fabs
          (fma
           (- 0.5 (* 0.5 (cos (* 2.0 t_1))))
           (* (cos phi2) (cos phi1))
           (- 0.5 (* 0.5 (cos (* 2.0 (* 0.5 (- phi1 phi2))))))))))
   (if (<= (* 2.0 (atan2 (sqrt t_3) (sqrt (- 1.0 t_3)))) 0.2)
     (*
      R
      (*
       2.0
       (atan2
        (sqrt (+ t_2 (* (cos phi2) (pow (sin t_1) 2.0))))
        (sqrt
         (-
          (- 1.0 (- 0.5 (* 0.5 (cos (* 2.0 (* (- phi1 phi2) 0.5))))))
          (*
           (- 0.5 (* 0.5 (cos (* 2.0 (* (- lambda1 lambda2) 0.5)))))
           (cos phi2)))))))
     (* R (* 2.0 (atan2 (sqrt t_4) (sqrt (- 1.0 t_4))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(((lambda1 - lambda2) / 2.0));
	double t_1 = 0.5 * (lambda1 - lambda2);
	double t_2 = pow(sin(((phi1 - phi2) / 2.0)), 2.0);
	double t_3 = t_2 + (((cos(phi1) * cos(phi2)) * t_0) * t_0);
	double t_4 = fabs(fma((0.5 - (0.5 * cos((2.0 * t_1)))), (cos(phi2) * cos(phi1)), (0.5 - (0.5 * cos((2.0 * (0.5 * (phi1 - phi2))))))));
	double tmp;
	if ((2.0 * atan2(sqrt(t_3), sqrt((1.0 - t_3)))) <= 0.2) {
		tmp = R * (2.0 * atan2(sqrt((t_2 + (cos(phi2) * pow(sin(t_1), 2.0)))), sqrt(((1.0 - (0.5 - (0.5 * cos((2.0 * ((phi1 - phi2) * 0.5)))))) - ((0.5 - (0.5 * cos((2.0 * ((lambda1 - lambda2) * 0.5))))) * cos(phi2))))));
	} else {
		tmp = R * (2.0 * atan2(sqrt(t_4), sqrt((1.0 - t_4))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_1 = Float64(0.5 * Float64(lambda1 - lambda2))
	t_2 = sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0
	t_3 = Float64(t_2 + Float64(Float64(Float64(cos(phi1) * cos(phi2)) * t_0) * t_0))
	t_4 = abs(fma(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * t_1)))), Float64(cos(phi2) * cos(phi1)), Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(0.5 * Float64(phi1 - phi2))))))))
	tmp = 0.0
	if (Float64(2.0 * atan(sqrt(t_3), sqrt(Float64(1.0 - t_3)))) <= 0.2)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(Float64(t_2 + Float64(cos(phi2) * (sin(t_1) ^ 2.0)))), sqrt(Float64(Float64(1.0 - Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(Float64(phi1 - phi2) * 0.5)))))) - Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(Float64(lambda1 - lambda2) * 0.5))))) * cos(phi2)))))));
	else
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_4), sqrt(Float64(1.0 - t_4)))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 + N[(N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Abs[N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] + N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(0.5 * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(2.0 * N[ArcTan[N[Sqrt[t$95$3], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$3), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.2], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(t$95$2 + N[(N[Cos[phi2], $MachinePrecision] * N[Power[N[Sin[t$95$1], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[(1.0 - N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(N[(phi1 - phi2), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(N[(lambda1 - lambda2), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$4], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$4), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 2 binary64) (atan2.f64 (sqrt.f64 (+.f64 (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) #s(literal 2 binary64))) #s(literal 2 binary64)) (*.f64 (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))))) (sqrt.f64 (-.f64 #s(literal 1 binary64) (+.f64 (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) #s(literal 2 binary64))) #s(literal 2 binary64)) (*.f64 (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64))))))))) < 0.20000000000000001

   1. Initial program 62.0%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 52.8

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 52.8%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_2 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}\right)}}\right) \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}\right) \]

      6. lower--.f64 50.7

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 50.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_2 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}\right)}}\right) \]

   9. Applied rewrites 50.8%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\color{blue}{\sqrt{\left(1 - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)\right) - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right) \cdot \cos \phi_2}}}\right) \]

   if 0.20000000000000001 < (*.f64 #s(literal 2 binary64) (atan2.f64 (sqrt.f64 (+.f64 (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) #s(literal 2 binary64))) #s(literal 2 binary64)) (*.f64 (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))))) (sqrt.f64 (-.f64 #s(literal 1 binary64) (+.f64 (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) #s(literal 2 binary64))) #s(literal 2 binary64)) (*.f64 (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))))))))

   1. Initial program 62.0%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 57.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\left|\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)\right|}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 57.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left|\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)\right|}}{\sqrt{1 - \color{blue}{\left|\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)\right|}}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 62.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := \cos \phi_2 \cdot \cos \phi_1\\ t_1 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_2 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\\ t_3 := 0.5 \cdot \left(\lambda_1 - \lambda_2\right)\\ t_4 := 0.5 - 0.5 \cdot \cos \left(2 \cdot t\_3\right)\\ t_5 := t\_2 + \cos \phi_1 \cdot {\sin t\_3}^{2}\\ t_6 := 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\\ \mathbf{if}\;t\_2 + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_1\right) \cdot t\_1 \leq 0.05:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_5}}{\sqrt{1 - t\_5}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(R \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(t\_4, t\_0, 0.5 - t\_6\right)}}{\sqrt{\left(0.5 + t\_6\right) - t\_4 \cdot t\_0}}\right) \cdot 2\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi2) (cos phi1)))
        (t_1 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_2 (pow (sin (/ (- phi1 phi2) 2.0)) 2.0))
        (t_3 (* 0.5 (- lambda1 lambda2)))
        (t_4 (- 0.5 (* 0.5 (cos (* 2.0 t_3)))))
        (t_5 (+ t_2 (* (cos phi1) (pow (sin t_3) 2.0))))
        (t_6 (* 0.5 (cos (* 2.0 (* 0.5 (- phi1 phi2)))))))
   (if (<= (+ t_2 (* (* (* (cos phi1) (cos phi2)) t_1) t_1)) 0.05)
     (* R (* 2.0 (atan2 (sqrt t_5) (sqrt (- 1.0 t_5)))))
     (*
      (*
       R
       (atan2
        (sqrt (fma t_4 t_0 (- 0.5 t_6)))
        (sqrt (- (+ 0.5 t_6) (* t_4 t_0)))))
      2.0))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi2) * cos(phi1);
	double t_1 = sin(((lambda1 - lambda2) / 2.0));
	double t_2 = pow(sin(((phi1 - phi2) / 2.0)), 2.0);
	double t_3 = 0.5 * (lambda1 - lambda2);
	double t_4 = 0.5 - (0.5 * cos((2.0 * t_3)));
	double t_5 = t_2 + (cos(phi1) * pow(sin(t_3), 2.0));
	double t_6 = 0.5 * cos((2.0 * (0.5 * (phi1 - phi2))));
	double tmp;
	if ((t_2 + (((cos(phi1) * cos(phi2)) * t_1) * t_1)) <= 0.05) {
		tmp = R * (2.0 * atan2(sqrt(t_5), sqrt((1.0 - t_5))));
	} else {
		tmp = (R * atan2(sqrt(fma(t_4, t_0, (0.5 - t_6))), sqrt(((0.5 + t_6) - (t_4 * t_0))))) * 2.0;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi2) * cos(phi1))
	t_1 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_2 = sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0
	t_3 = Float64(0.5 * Float64(lambda1 - lambda2))
	t_4 = Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * t_3))))
	t_5 = Float64(t_2 + Float64(cos(phi1) * (sin(t_3) ^ 2.0)))
	t_6 = Float64(0.5 * cos(Float64(2.0 * Float64(0.5 * Float64(phi1 - phi2)))))
	tmp = 0.0
	if (Float64(t_2 + Float64(Float64(Float64(cos(phi1) * cos(phi2)) * t_1) * t_1)) <= 0.05)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_5), sqrt(Float64(1.0 - t_5)))));
	else
		tmp = Float64(Float64(R * atan(sqrt(fma(t_4, t_0, Float64(0.5 - t_6))), sqrt(Float64(Float64(0.5 + t_6) - Float64(t_4 * t_0))))) * 2.0);
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * t$95$3), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$2 + N[(N[Cos[phi1], $MachinePrecision] * N[Power[N[Sin[t$95$3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(0.5 * N[Cos[N[(2.0 * N[(0.5 * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$2 + N[(N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], 0.05], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$5], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(R * N[ArcTan[N[Sqrt[N[(t$95$4 * t$95$0 + N[(0.5 - t$95$6), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[(0.5 + t$95$6), $MachinePrecision] - N[(t$95$4 * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) #s(literal 2 binary64))) #s(literal 2 binary64)) (*.f64 (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64))))) < 0.050000000000000003

   1. Initial program 62.0%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 53.2

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 53.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}\right)}}\right) \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}\right) \]

      6. lower--.f64 51.1

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 51.1%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}\right)}}\right) \]

   if 0.050000000000000003 < (+.f64 (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) #s(literal 2 binary64))) #s(literal 2 binary64)) (*.f64 (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))))

   1. Initial program 62.0%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 57.1%

      \[\leadsto \color{blue}{\left(R \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right) - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}}\right) \cdot 2} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 22: 62.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := 0.5 \cdot \cos \left(\phi_1 - \phi_2\right)\\ t_1 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_2 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\\ t_3 := \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\\ t_4 := t\_2 + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\\ \mathbf{if}\;t\_2 + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_1\right) \cdot t\_1 \leq 0.05:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_4}}{\sqrt{1 - t\_4}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(R \cdot \tan^{-1}_* \frac{\sqrt{\left(0.5 + t\_3\right) - t\_0}}{\sqrt{\left(0.5 + t\_0\right) - t\_3}}\right) \cdot 2\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* 0.5 (cos (- phi1 phi2))))
        (t_1 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_2 (pow (sin (/ (- phi1 phi2) 2.0)) 2.0))
        (t_3
         (*
          (cos phi1)
          (* (cos phi2) (- 0.5 (* 0.5 (cos (- lambda1 lambda2)))))))
        (t_4
         (+ t_2 (* (cos phi1) (pow (sin (* 0.5 (- lambda1 lambda2))) 2.0)))))
   (if (<= (+ t_2 (* (* (* (cos phi1) (cos phi2)) t_1) t_1)) 0.05)
     (* R (* 2.0 (atan2 (sqrt t_4) (sqrt (- 1.0 t_4)))))
     (*
      (* R (atan2 (sqrt (- (+ 0.5 t_3) t_0)) (sqrt (- (+ 0.5 t_0) t_3))))
      2.0))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = 0.5 * cos((phi1 - phi2));
	double t_1 = sin(((lambda1 - lambda2) / 2.0));
	double t_2 = pow(sin(((phi1 - phi2) / 2.0)), 2.0);
	double t_3 = cos(phi1) * (cos(phi2) * (0.5 - (0.5 * cos((lambda1 - lambda2)))));
	double t_4 = t_2 + (cos(phi1) * pow(sin((0.5 * (lambda1 - lambda2))), 2.0));
	double tmp;
	if ((t_2 + (((cos(phi1) * cos(phi2)) * t_1) * t_1)) <= 0.05) {
		tmp = R * (2.0 * atan2(sqrt(t_4), sqrt((1.0 - t_4))));
	} else {
		tmp = (R * atan2(sqrt(((0.5 + t_3) - t_0)), sqrt(((0.5 + t_0) - t_3)))) * 2.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(r, lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_0 = 0.5d0 * cos((phi1 - phi2))
    t_1 = sin(((lambda1 - lambda2) / 2.0d0))
    t_2 = sin(((phi1 - phi2) / 2.0d0)) ** 2.0d0
    t_3 = cos(phi1) * (cos(phi2) * (0.5d0 - (0.5d0 * cos((lambda1 - lambda2)))))
    t_4 = t_2 + (cos(phi1) * (sin((0.5d0 * (lambda1 - lambda2))) ** 2.0d0))
    if ((t_2 + (((cos(phi1) * cos(phi2)) * t_1) * t_1)) <= 0.05d0) then
        tmp = r * (2.0d0 * atan2(sqrt(t_4), sqrt((1.0d0 - t_4))))
    else
        tmp = (r * atan2(sqrt(((0.5d0 + t_3) - t_0)), sqrt(((0.5d0 + t_0) - t_3)))) * 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = 0.5 * Math.cos((phi1 - phi2));
	double t_1 = Math.sin(((lambda1 - lambda2) / 2.0));
	double t_2 = Math.pow(Math.sin(((phi1 - phi2) / 2.0)), 2.0);
	double t_3 = Math.cos(phi1) * (Math.cos(phi2) * (0.5 - (0.5 * Math.cos((lambda1 - lambda2)))));
	double t_4 = t_2 + (Math.cos(phi1) * Math.pow(Math.sin((0.5 * (lambda1 - lambda2))), 2.0));
	double tmp;
	if ((t_2 + (((Math.cos(phi1) * Math.cos(phi2)) * t_1) * t_1)) <= 0.05) {
		tmp = R * (2.0 * Math.atan2(Math.sqrt(t_4), Math.sqrt((1.0 - t_4))));
	} else {
		tmp = (R * Math.atan2(Math.sqrt(((0.5 + t_3) - t_0)), Math.sqrt(((0.5 + t_0) - t_3)))) * 2.0;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = 0.5 * math.cos((phi1 - phi2))
	t_1 = math.sin(((lambda1 - lambda2) / 2.0))
	t_2 = math.pow(math.sin(((phi1 - phi2) / 2.0)), 2.0)
	t_3 = math.cos(phi1) * (math.cos(phi2) * (0.5 - (0.5 * math.cos((lambda1 - lambda2)))))
	t_4 = t_2 + (math.cos(phi1) * math.pow(math.sin((0.5 * (lambda1 - lambda2))), 2.0))
	tmp = 0
	if (t_2 + (((math.cos(phi1) * math.cos(phi2)) * t_1) * t_1)) <= 0.05:
		tmp = R * (2.0 * math.atan2(math.sqrt(t_4), math.sqrt((1.0 - t_4))))
	else:
		tmp = (R * math.atan2(math.sqrt(((0.5 + t_3) - t_0)), math.sqrt(((0.5 + t_0) - t_3)))) * 2.0
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(0.5 * cos(Float64(phi1 - phi2)))
	t_1 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_2 = sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0
	t_3 = Float64(cos(phi1) * Float64(cos(phi2) * Float64(0.5 - Float64(0.5 * cos(Float64(lambda1 - lambda2))))))
	t_4 = Float64(t_2 + Float64(cos(phi1) * (sin(Float64(0.5 * Float64(lambda1 - lambda2))) ^ 2.0)))
	tmp = 0.0
	if (Float64(t_2 + Float64(Float64(Float64(cos(phi1) * cos(phi2)) * t_1) * t_1)) <= 0.05)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_4), sqrt(Float64(1.0 - t_4)))));
	else
		tmp = Float64(Float64(R * atan(sqrt(Float64(Float64(0.5 + t_3) - t_0)), sqrt(Float64(Float64(0.5 + t_0) - t_3)))) * 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = 0.5 * cos((phi1 - phi2));
	t_1 = sin(((lambda1 - lambda2) / 2.0));
	t_2 = sin(((phi1 - phi2) / 2.0)) ^ 2.0;
	t_3 = cos(phi1) * (cos(phi2) * (0.5 - (0.5 * cos((lambda1 - lambda2)))));
	t_4 = t_2 + (cos(phi1) * (sin((0.5 * (lambda1 - lambda2))) ^ 2.0));
	tmp = 0.0;
	if ((t_2 + (((cos(phi1) * cos(phi2)) * t_1) * t_1)) <= 0.05)
		tmp = R * (2.0 * atan2(sqrt(t_4), sqrt((1.0 - t_4))));
	else
		tmp = (R * atan2(sqrt(((0.5 + t_3) - t_0)), sqrt(((0.5 + t_0) - t_3)))) * 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(0.5 * N[Cos[N[(phi1 - phi2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$2 + N[(N[Cos[phi1], $MachinePrecision] * N[Power[N[Sin[N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$2 + N[(N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], 0.05], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$4], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$4), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(R * N[ArcTan[N[Sqrt[N[(N[(0.5 + t$95$3), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[(0.5 + t$95$0), $MachinePrecision] - t$95$3), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) #s(literal 2 binary64))) #s(literal 2 binary64)) (*.f64 (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64))))) < 0.050000000000000003

   1. Initial program 62.0%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 53.2

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 53.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}\right)}}\right) \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}\right) \]

      6. lower--.f64 51.1

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 51.1%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}\right)}}\right) \]

   if 0.050000000000000003 < (+.f64 (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) #s(literal 2 binary64))) #s(literal 2 binary64)) (*.f64 (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))))

   1. Initial program 62.0%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 57.1%

      \[\leadsto \color{blue}{\left(R \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right) - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}}\right) \cdot 2} \]

   4. Taylor expanded in lambda1 around 0

      \[\leadsto \left(R \cdot \color{blue}{\tan^{-1}_* \frac{\sqrt{\left(\frac{1}{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right) - \frac{1}{2} \cdot \cos \left(\phi_1 - \phi_2\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\phi_1 - \phi_2\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}}\right) \cdot 2 \]
   5. Step-by-step derivation

      1. lower-atan2.f64 N/A

         \[\leadsto \left(R \cdot \tan^{-1}_* \frac{\sqrt{\left(\frac{1}{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right) - \frac{1}{2} \cdot \cos \left(\phi_1 - \phi_2\right)}}{\color{blue}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\phi_1 - \phi_2\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}}\right) \cdot 2 \]

   6. Applied rewrites 57.0%

      \[\leadsto \left(R \cdot \color{blue}{\tan^{-1}_* \frac{\sqrt{\left(0.5 + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right) - 0.5 \cdot \cos \left(\phi_1 - \phi_2\right)}}{\sqrt{\left(0.5 + 0.5 \cdot \cos \left(\phi_1 - \phi_2\right)\right) - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}}\right) \cdot 2 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 23: 61.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := \cos \left(\left(\phi_2 - \phi_1\right) \cdot 0.5\right)\\ R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_0\right) \cdot t\_0}}{\sqrt{\mathsf{fma}\left(t\_1, t\_1, \left(-\cos \phi_2 \cdot \cos \phi_1\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\right)}}\right) \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_1 (cos (* (- phi2 phi1) 0.5))))
   (*
    R
    (*
     2.0
     (atan2
      (sqrt
       (+
        (pow (sin (/ (- phi1 phi2) 2.0)) 2.0)
        (* (* (* (cos phi1) (cos phi2)) t_0) t_0)))
      (sqrt
       (fma
        t_1
        t_1
        (*
         (- (* (cos phi2) (cos phi1)))
         (- 0.5 (* 0.5 (cos (* 2.0 (* 0.5 (- lambda1 lambda2))))))))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(((lambda1 - lambda2) / 2.0));
	double t_1 = cos(((phi2 - phi1) * 0.5));
	return R * (2.0 * atan2(sqrt((pow(sin(((phi1 - phi2) / 2.0)), 2.0) + (((cos(phi1) * cos(phi2)) * t_0) * t_0))), sqrt(fma(t_1, t_1, (-(cos(phi2) * cos(phi1)) * (0.5 - (0.5 * cos((2.0 * (0.5 * (lambda1 - lambda2)))))))))));
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_1 = cos(Float64(Float64(phi2 - phi1) * 0.5))
	return Float64(R * Float64(2.0 * atan(sqrt(Float64((sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0) + Float64(Float64(Float64(cos(phi1) * cos(phi2)) * t_0) * t_0))), sqrt(fma(t_1, t_1, Float64(Float64(-Float64(cos(phi2) * cos(phi1))) * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(0.5 * Float64(lambda1 - lambda2))))))))))))
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(N[(phi2 - phi1), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]}, N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[(N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(t$95$1 * t$95$1 + N[((-N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]) * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 62.0%

   \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

3. Applied rewrites 62.1%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \left(\left(\phi_2 - \phi_1\right) \cdot 0.5\right), \cos \left(\left(\phi_2 - \phi_1\right) \cdot 0.5\right), \left(-\cos \phi_2 \cdot \cos \phi_1\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\right)}}}\right) \]
4. Add Preprocessing

Alternative 24: 61.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_0\right) \cdot t\_0}}{\sqrt{-\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, -\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)\right)}}\right) \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (/ (- lambda1 lambda2) 2.0))))
   (*
    R
    (*
     2.0
     (atan2
      (sqrt
       (+
        (pow (sin (/ (- phi1 phi2) 2.0)) 2.0)
        (* (* (* (cos phi1) (cos phi2)) t_0) t_0)))
      (sqrt
       (-
        (fma
         (- 0.5 (* 0.5 (cos (* 2.0 (* 0.5 (- lambda1 lambda2))))))
         (* (cos phi2) (cos phi1))
         (- (+ 0.5 (* 0.5 (cos (* 2.0 (* 0.5 (- phi1 phi2)))))))))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(((lambda1 - lambda2) / 2.0));
	return R * (2.0 * atan2(sqrt((pow(sin(((phi1 - phi2) / 2.0)), 2.0) + (((cos(phi1) * cos(phi2)) * t_0) * t_0))), sqrt(-fma((0.5 - (0.5 * cos((2.0 * (0.5 * (lambda1 - lambda2)))))), (cos(phi2) * cos(phi1)), -(0.5 + (0.5 * cos((2.0 * (0.5 * (phi1 - phi2))))))))));
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	return Float64(R * Float64(2.0 * atan(sqrt(Float64((sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0) + Float64(Float64(Float64(cos(phi1) * cos(phi2)) * t_0) * t_0))), sqrt(Float64(-fma(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(0.5 * Float64(lambda1 - lambda2)))))), Float64(cos(phi2) * cos(phi1)), Float64(-Float64(0.5 + Float64(0.5 * cos(Float64(2.0 * Float64(0.5 * Float64(phi1 - phi2)))))))))))))
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[(N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[(-N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] + (-N[(0.5 + N[(0.5 * N[Cos[N[(2.0 * N[(0.5 * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision])), $MachinePrecision])], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 62.0%

   \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

3. Applied rewrites 62.1%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\color{blue}{-\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, -\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)\right)}}}\right) \]
4. Add Preprocessing

Alternative 25: 61.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_0\right) \cdot t\_0}}{\sqrt{\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right) - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}}\right) \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (/ (- lambda1 lambda2) 2.0))))
   (*
    R
    (*
     2.0
     (atan2
      (sqrt
       (+
        (pow (sin (/ (- phi1 phi2) 2.0)) 2.0)
        (* (* (* (cos phi1) (cos phi2)) t_0) t_0)))
      (sqrt
       (-
        (+ 0.5 (* 0.5 (cos (* 2.0 (* 0.5 (- phi1 phi2))))))
        (*
         (- 0.5 (* 0.5 (cos (* 2.0 (* 0.5 (- lambda1 lambda2))))))
         (* (cos phi2) (cos phi1))))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(((lambda1 - lambda2) / 2.0));
	return R * (2.0 * atan2(sqrt((pow(sin(((phi1 - phi2) / 2.0)), 2.0) + (((cos(phi1) * cos(phi2)) * t_0) * t_0))), sqrt(((0.5 + (0.5 * cos((2.0 * (0.5 * (phi1 - phi2)))))) - ((0.5 - (0.5 * cos((2.0 * (0.5 * (lambda1 - lambda2)))))) * (cos(phi2) * cos(phi1)))))));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(r, lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    t_0 = sin(((lambda1 - lambda2) / 2.0d0))
    code = r * (2.0d0 * atan2(sqrt(((sin(((phi1 - phi2) / 2.0d0)) ** 2.0d0) + (((cos(phi1) * cos(phi2)) * t_0) * t_0))), sqrt(((0.5d0 + (0.5d0 * cos((2.0d0 * (0.5d0 * (phi1 - phi2)))))) - ((0.5d0 - (0.5d0 * cos((2.0d0 * (0.5d0 * (lambda1 - lambda2)))))) * (cos(phi2) * cos(phi1)))))))
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.sin(((lambda1 - lambda2) / 2.0));
	return R * (2.0 * Math.atan2(Math.sqrt((Math.pow(Math.sin(((phi1 - phi2) / 2.0)), 2.0) + (((Math.cos(phi1) * Math.cos(phi2)) * t_0) * t_0))), Math.sqrt(((0.5 + (0.5 * Math.cos((2.0 * (0.5 * (phi1 - phi2)))))) - ((0.5 - (0.5 * Math.cos((2.0 * (0.5 * (lambda1 - lambda2)))))) * (Math.cos(phi2) * Math.cos(phi1)))))));
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.sin(((lambda1 - lambda2) / 2.0))
	return R * (2.0 * math.atan2(math.sqrt((math.pow(math.sin(((phi1 - phi2) / 2.0)), 2.0) + (((math.cos(phi1) * math.cos(phi2)) * t_0) * t_0))), math.sqrt(((0.5 + (0.5 * math.cos((2.0 * (0.5 * (phi1 - phi2)))))) - ((0.5 - (0.5 * math.cos((2.0 * (0.5 * (lambda1 - lambda2)))))) * (math.cos(phi2) * math.cos(phi1)))))))

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	return Float64(R * Float64(2.0 * atan(sqrt(Float64((sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0) + Float64(Float64(Float64(cos(phi1) * cos(phi2)) * t_0) * t_0))), sqrt(Float64(Float64(0.5 + Float64(0.5 * cos(Float64(2.0 * Float64(0.5 * Float64(phi1 - phi2)))))) - Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(0.5 * Float64(lambda1 - lambda2)))))) * Float64(cos(phi2) * cos(phi1))))))))
end

MATLAB

function tmp = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(((lambda1 - lambda2) / 2.0));
	tmp = R * (2.0 * atan2(sqrt(((sin(((phi1 - phi2) / 2.0)) ^ 2.0) + (((cos(phi1) * cos(phi2)) * t_0) * t_0))), sqrt(((0.5 + (0.5 * cos((2.0 * (0.5 * (phi1 - phi2)))))) - ((0.5 - (0.5 * cos((2.0 * (0.5 * (lambda1 - lambda2)))))) * (cos(phi2) * cos(phi1)))))));
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[(N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[(0.5 + N[(0.5 * N[Cos[N[(2.0 * N[(0.5 * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 62.0%

   \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

3. Applied rewrites 62.1%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\color{blue}{\sqrt{\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right) - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}}}\right) \]
4. Add Preprocessing

Alternative 26: 61.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} t_0 := 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\\ t_1 := \mathsf{fma}\left(\cos \phi_2, t\_0, {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)\\ t_2 := \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\phi_2 \cdot 0.5\right)\right)\right) + t\_0 \cdot \cos \phi_2\\ \mathbf{if}\;\phi_2 \leq -1.15 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_2}}{\sqrt{1 - t\_2}}\right)\\ \mathbf{elif}\;\phi_2 \leq 5 \cdot 10^{-27}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(t\_0, \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_1}}{\sqrt{1 - t\_1}}\right)\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (- 0.5 (* 0.5 (cos (* 2.0 (* (- lambda1 lambda2) 0.5))))))
        (t_1 (fma (cos phi2) t_0 (pow (sin (* 0.5 phi2)) 2.0)))
        (t_2
         (+ (- 0.5 (* 0.5 (cos (* 2.0 (* phi2 0.5))))) (* t_0 (cos phi2)))))
   (if (<= phi2 -1.15e-5)
     (* R (* 2.0 (atan2 (sqrt t_2) (sqrt (- 1.0 t_2)))))
     (if (<= phi2 5e-27)
       (*
        R
        (*
         2.0
         (atan2
          (sqrt
           (fma
            (cos phi1)
            (pow (sin (* 0.5 (- lambda1 lambda2))) 2.0)
            (pow (sin (* 0.5 phi1)) 2.0)))
          (sqrt
           (-
            1.0
            (fma
             t_0
             (cos phi1)
             (- 0.5 (* 0.5 (cos (* 2.0 (* 0.5 phi1)))))))))))
       (* R (* 2.0 (atan2 (sqrt t_1) (sqrt (- 1.0 t_1)))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = 0.5 - (0.5 * cos((2.0 * ((lambda1 - lambda2) * 0.5))));
	double t_1 = fma(cos(phi2), t_0, pow(sin((0.5 * phi2)), 2.0));
	double t_2 = (0.5 - (0.5 * cos((2.0 * (phi2 * 0.5))))) + (t_0 * cos(phi2));
	double tmp;
	if (phi2 <= -1.15e-5) {
		tmp = R * (2.0 * atan2(sqrt(t_2), sqrt((1.0 - t_2))));
	} else if (phi2 <= 5e-27) {
		tmp = R * (2.0 * atan2(sqrt(fma(cos(phi1), pow(sin((0.5 * (lambda1 - lambda2))), 2.0), pow(sin((0.5 * phi1)), 2.0))), sqrt((1.0 - fma(t_0, cos(phi1), (0.5 - (0.5 * cos((2.0 * (0.5 * phi1))))))))));
	} else {
		tmp = R * (2.0 * atan2(sqrt(t_1), sqrt((1.0 - t_1))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(Float64(lambda1 - lambda2) * 0.5)))))
	t_1 = fma(cos(phi2), t_0, (sin(Float64(0.5 * phi2)) ^ 2.0))
	t_2 = Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(phi2 * 0.5))))) + Float64(t_0 * cos(phi2)))
	tmp = 0.0
	if (phi2 <= -1.15e-5)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_2), sqrt(Float64(1.0 - t_2)))));
	elseif (phi2 <= 5e-27)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(fma(cos(phi1), (sin(Float64(0.5 * Float64(lambda1 - lambda2))) ^ 2.0), (sin(Float64(0.5 * phi1)) ^ 2.0))), sqrt(Float64(1.0 - fma(t_0, cos(phi1), Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(0.5 * phi1)))))))))));
	else
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_1), sqrt(Float64(1.0 - t_1)))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(N[(lambda1 - lambda2), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[phi2], $MachinePrecision] * t$95$0 + N[Power[N[Sin[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(phi2 * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -1.15e-5], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$2], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$2), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 5e-27], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[Cos[phi1], $MachinePrecision] * N[Power[N[Sin[N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(t$95$0 * N[Cos[phi1], $MachinePrecision] + N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(0.5 * phi1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$1], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -1.15e-5

   1. Initial program 62.0%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 61.1%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 62.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 62.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}}\right) \]

   9. Applied rewrites 77.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right)}}\right) \]
   10. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       7. mult-flip-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       8. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       9. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       10. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       11. lift-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       12. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       13. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       14. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       15. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       17. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       18. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       19. mult-flip-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       20. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       21. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       22. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       23. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       24. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       25. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   11. Applied rewrites 78.2%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   12. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       7. mult-flip-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       8. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       9. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       10. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       11. lift-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       12. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       13. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       14. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       15. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       17. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       18. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       19. mult-flip-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       20. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       21. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       22. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       23. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       24. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       25. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   13. Applied rewrites 98.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   14. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}}}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   15. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \color{blue}{{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   16. Applied rewrites 57.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   17. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}}\right) \]
   18. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, \color{blue}{{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

   19. Applied rewrites 56.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)}}}\right) \]

   21. Applied rewrites 42.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\phi_2 \cdot 0.5\right)\right)\right) + \color{blue}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right) \cdot \cos \phi_2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)}}\right) \]

   23. Applied rewrites 42.4%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\phi_2 \cdot 0.5\right)\right)\right) + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right) \cdot \cos \phi_2}}{\sqrt{1 - \left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\phi_2 \cdot 0.5\right)\right)\right) + \color{blue}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right) \cdot \cos \phi_2}\right)}}\right) \]

   if -1.15e-5 < phi2 < 5.0000000000000002e-27

   1. Initial program 62.0%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-*.f64 46.7

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 46.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   6. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      9. lower-*.f64 46.8

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 46.8%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]

   9. Applied rewrites 46.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\color{blue}{\sqrt{1 - \mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}}\right) \]

   if 5.0000000000000002e-27 < phi2

   1. Initial program 62.0%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 61.1%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 62.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 62.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}}\right) \]

   9. Applied rewrites 77.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right)}}\right) \]
   10. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       7. mult-flip-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       8. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       9. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       10. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       11. lift-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       12. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       13. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       14. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       15. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       17. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       18. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       19. mult-flip-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       20. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       21. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       22. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       23. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       24. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       25. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   11. Applied rewrites 78.2%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   12. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       7. mult-flip-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       8. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       9. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       10. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       11. lift-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       12. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       13. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       14. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       15. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       17. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       18. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       19. mult-flip-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       20. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       21. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       22. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       23. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       24. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       25. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   13. Applied rewrites 98.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   14. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}}}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   15. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \color{blue}{{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   16. Applied rewrites 57.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   17. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}}\right) \]
   18. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, \color{blue}{{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

   19. Applied rewrites 56.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)}}}\right) \]
   20. Step-by-step derivation

       1. lift-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

   21. Applied rewrites 44.1%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, 0.5 - \color{blue}{0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)}, {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)}}\right) \]
   22. Step-by-step derivation

       1. lift-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

   23. Applied rewrites 43.8%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right), {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, 0.5 - \color{blue}{0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)}, {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)}}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 27: 61.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} t_0 := 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\phi_2 \cdot 0.5\right)\right)\\ t_1 := 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\\ t_2 := \mathsf{fma}\left(t\_1, \cos \phi_2, t\_0\right)\\ t_3 := t\_0 + t\_1 \cdot \cos \phi_2\\ \mathbf{if}\;\phi_2 \leq -1.15 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_3}}{\sqrt{1 - t\_3}}\right)\\ \mathbf{elif}\;\phi_2 \leq 5 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(t\_1, \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\tan^{-1}_* \frac{\sqrt{t\_2}}{\sqrt{1 - t\_2}} \cdot 2\right) \cdot R\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (- 0.5 (* 0.5 (cos (* 2.0 (* phi2 0.5))))))
        (t_1 (- 0.5 (* 0.5 (cos (* 2.0 (* (- lambda1 lambda2) 0.5))))))
        (t_2 (fma t_1 (cos phi2) t_0))
        (t_3 (+ t_0 (* t_1 (cos phi2)))))
   (if (<= phi2 -1.15e-5)
     (* R (* 2.0 (atan2 (sqrt t_3) (sqrt (- 1.0 t_3)))))
     (if (<= phi2 5e-5)
       (*
        R
        (*
         2.0
         (atan2
          (sqrt
           (fma
            (cos phi1)
            (pow (sin (* 0.5 (- lambda1 lambda2))) 2.0)
            (pow (sin (* 0.5 phi1)) 2.0)))
          (sqrt
           (-
            1.0
            (fma
             t_1
             (cos phi1)
             (- 0.5 (* 0.5 (cos (* 2.0 (* 0.5 phi1)))))))))))
       (* (* (atan2 (sqrt t_2) (sqrt (- 1.0 t_2))) 2.0) R)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = 0.5 - (0.5 * cos((2.0 * (phi2 * 0.5))));
	double t_1 = 0.5 - (0.5 * cos((2.0 * ((lambda1 - lambda2) * 0.5))));
	double t_2 = fma(t_1, cos(phi2), t_0);
	double t_3 = t_0 + (t_1 * cos(phi2));
	double tmp;
	if (phi2 <= -1.15e-5) {
		tmp = R * (2.0 * atan2(sqrt(t_3), sqrt((1.0 - t_3))));
	} else if (phi2 <= 5e-5) {
		tmp = R * (2.0 * atan2(sqrt(fma(cos(phi1), pow(sin((0.5 * (lambda1 - lambda2))), 2.0), pow(sin((0.5 * phi1)), 2.0))), sqrt((1.0 - fma(t_1, cos(phi1), (0.5 - (0.5 * cos((2.0 * (0.5 * phi1))))))))));
	} else {
		tmp = (atan2(sqrt(t_2), sqrt((1.0 - t_2))) * 2.0) * R;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(phi2 * 0.5)))))
	t_1 = Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(Float64(lambda1 - lambda2) * 0.5)))))
	t_2 = fma(t_1, cos(phi2), t_0)
	t_3 = Float64(t_0 + Float64(t_1 * cos(phi2)))
	tmp = 0.0
	if (phi2 <= -1.15e-5)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_3), sqrt(Float64(1.0 - t_3)))));
	elseif (phi2 <= 5e-5)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(fma(cos(phi1), (sin(Float64(0.5 * Float64(lambda1 - lambda2))) ^ 2.0), (sin(Float64(0.5 * phi1)) ^ 2.0))), sqrt(Float64(1.0 - fma(t_1, cos(phi1), Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(0.5 * phi1)))))))))));
	else
		tmp = Float64(Float64(atan(sqrt(t_2), sqrt(Float64(1.0 - t_2))) * 2.0) * R);
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(phi2 * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(N[(lambda1 - lambda2), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[Cos[phi2], $MachinePrecision] + t$95$0), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$0 + N[(t$95$1 * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -1.15e-5], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$3], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$3), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 5e-5], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[Cos[phi1], $MachinePrecision] * N[Power[N[Sin[N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(t$95$1 * N[Cos[phi1], $MachinePrecision] + N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(0.5 * phi1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[ArcTan[N[Sqrt[t$95$2], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$2), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * R), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -1.15e-5

   1. Initial program 62.0%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 61.1%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 62.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 62.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}}\right) \]

   9. Applied rewrites 77.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right)}}\right) \]
   10. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       7. mult-flip-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       8. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       9. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       10. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       11. lift-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       12. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       13. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       14. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       15. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       17. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       18. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       19. mult-flip-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       20. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       21. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       22. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       23. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       24. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       25. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   11. Applied rewrites 78.2%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   12. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       7. mult-flip-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       8. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       9. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       10. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       11. lift-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       12. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       13. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       14. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       15. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       17. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       18. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       19. mult-flip-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       20. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       21. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       22. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       23. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       24. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       25. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   13. Applied rewrites 98.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   14. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}}}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   15. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \color{blue}{{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   16. Applied rewrites 57.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   17. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}}\right) \]
   18. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, \color{blue}{{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

   19. Applied rewrites 56.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)}}}\right) \]

   21. Applied rewrites 42.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\phi_2 \cdot 0.5\right)\right)\right) + \color{blue}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right) \cdot \cos \phi_2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)}}\right) \]

   23. Applied rewrites 42.4%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\phi_2 \cdot 0.5\right)\right)\right) + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right) \cdot \cos \phi_2}}{\sqrt{1 - \left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\phi_2 \cdot 0.5\right)\right)\right) + \color{blue}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right) \cdot \cos \phi_2}\right)}}\right) \]

   if -1.15e-5 < phi2 < 5.00000000000000024e-5

   1. Initial program 62.0%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-*.f64 46.7

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 46.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   6. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      9. lower-*.f64 46.8

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 46.8%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]

   9. Applied rewrites 46.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\color{blue}{\sqrt{1 - \mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}}\right) \]

   if 5.00000000000000024e-5 < phi2

   1. Initial program 62.0%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 61.1%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 62.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 62.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}}\right) \]

   9. Applied rewrites 77.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right)}}\right) \]
   10. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       7. mult-flip-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       8. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       9. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       10. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       11. lift-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       12. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       13. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       14. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       15. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       17. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       18. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       19. mult-flip-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       20. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       21. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       22. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       23. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       24. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       25. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   11. Applied rewrites 78.2%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   12. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       7. mult-flip-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       8. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       9. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       10. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       11. lift-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       12. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       13. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       14. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       15. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       17. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       18. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       19. mult-flip-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       20. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       21. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       22. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       23. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       24. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       25. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   13. Applied rewrites 98.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   14. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}}}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   15. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \color{blue}{{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   16. Applied rewrites 57.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   17. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}}\right) \]
   18. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, \color{blue}{{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

   19. Applied rewrites 56.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)}}}\right) \]

   21. Applied rewrites 42.4%

       \[\leadsto \color{blue}{\left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right), \cos \phi_2, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\phi_2 \cdot 0.5\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right), \cos \phi_2, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\phi_2 \cdot 0.5\right)\right)\right)}} \cdot 2\right) \cdot R} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 28: 60.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} t_0 := 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\phi_2 \cdot 0.5\right)\right)\\ t_1 := 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\\ t_2 := \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)\\ t_3 := \mathsf{fma}\left(t\_1, \cos \phi_2, t\_0\right)\\ t_4 := t\_0 + t\_1 \cdot \cos \phi_2\\ \mathbf{if}\;\phi_2 \leq -1.15 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_4}}{\sqrt{1 - t\_4}}\right)\\ \mathbf{elif}\;\phi_2 \leq 5 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_2}}{\sqrt{1 - t\_2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\tan^{-1}_* \frac{\sqrt{t\_3}}{\sqrt{1 - t\_3}} \cdot 2\right) \cdot R\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (- 0.5 (* 0.5 (cos (* 2.0 (* phi2 0.5))))))
        (t_1 (- 0.5 (* 0.5 (cos (* 2.0 (* (- lambda1 lambda2) 0.5))))))
        (t_2
         (fma
          (cos phi1)
          (pow (sin (* 0.5 (- lambda1 lambda2))) 2.0)
          (- 0.5 (* 0.5 (cos (* 2.0 (* 0.5 phi1)))))))
        (t_3 (fma t_1 (cos phi2) t_0))
        (t_4 (+ t_0 (* t_1 (cos phi2)))))
   (if (<= phi2 -1.15e-5)
     (* R (* 2.0 (atan2 (sqrt t_4) (sqrt (- 1.0 t_4)))))
     (if (<= phi2 5e-5)
       (* R (* 2.0 (atan2 (sqrt t_2) (sqrt (- 1.0 t_2)))))
       (* (* (atan2 (sqrt t_3) (sqrt (- 1.0 t_3))) 2.0) R)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = 0.5 - (0.5 * cos((2.0 * (phi2 * 0.5))));
	double t_1 = 0.5 - (0.5 * cos((2.0 * ((lambda1 - lambda2) * 0.5))));
	double t_2 = fma(cos(phi1), pow(sin((0.5 * (lambda1 - lambda2))), 2.0), (0.5 - (0.5 * cos((2.0 * (0.5 * phi1))))));
	double t_3 = fma(t_1, cos(phi2), t_0);
	double t_4 = t_0 + (t_1 * cos(phi2));
	double tmp;
	if (phi2 <= -1.15e-5) {
		tmp = R * (2.0 * atan2(sqrt(t_4), sqrt((1.0 - t_4))));
	} else if (phi2 <= 5e-5) {
		tmp = R * (2.0 * atan2(sqrt(t_2), sqrt((1.0 - t_2))));
	} else {
		tmp = (atan2(sqrt(t_3), sqrt((1.0 - t_3))) * 2.0) * R;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(phi2 * 0.5)))))
	t_1 = Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(Float64(lambda1 - lambda2) * 0.5)))))
	t_2 = fma(cos(phi1), (sin(Float64(0.5 * Float64(lambda1 - lambda2))) ^ 2.0), Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(0.5 * phi1))))))
	t_3 = fma(t_1, cos(phi2), t_0)
	t_4 = Float64(t_0 + Float64(t_1 * cos(phi2)))
	tmp = 0.0
	if (phi2 <= -1.15e-5)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_4), sqrt(Float64(1.0 - t_4)))));
	elseif (phi2 <= 5e-5)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_2), sqrt(Float64(1.0 - t_2)))));
	else
		tmp = Float64(Float64(atan(sqrt(t_3), sqrt(Float64(1.0 - t_3))) * 2.0) * R);
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(phi2 * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(N[(lambda1 - lambda2), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[phi1], $MachinePrecision] * N[Power[N[Sin[N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(0.5 * phi1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 * N[Cos[phi2], $MachinePrecision] + t$95$0), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$0 + N[(t$95$1 * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -1.15e-5], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$4], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$4), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 5e-5], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$2], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$2), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[ArcTan[N[Sqrt[t$95$3], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$3), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * R), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -1.15e-5

   1. Initial program 62.0%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 61.1%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 62.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 62.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}}\right) \]

   9. Applied rewrites 77.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right)}}\right) \]
   10. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       7. mult-flip-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       8. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       9. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       10. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       11. lift-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       12. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       13. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       14. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       15. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       17. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       18. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       19. mult-flip-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       20. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       21. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       22. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       23. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       24. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       25. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   11. Applied rewrites 78.2%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   12. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       7. mult-flip-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       8. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       9. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       10. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       11. lift-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       12. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       13. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       14. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       15. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       17. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       18. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       19. mult-flip-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       20. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       21. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       22. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       23. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       24. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       25. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   13. Applied rewrites 98.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   14. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}}}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   15. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \color{blue}{{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   16. Applied rewrites 57.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   17. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}}\right) \]
   18. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, \color{blue}{{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

   19. Applied rewrites 56.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)}}}\right) \]

   21. Applied rewrites 42.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\phi_2 \cdot 0.5\right)\right)\right) + \color{blue}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right) \cdot \cos \phi_2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)}}\right) \]

   23. Applied rewrites 42.4%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\phi_2 \cdot 0.5\right)\right)\right) + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right) \cdot \cos \phi_2}}{\sqrt{1 - \left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\phi_2 \cdot 0.5\right)\right)\right) + \color{blue}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right) \cdot \cos \phi_2}\right)}}\right) \]

   if -1.15e-5 < phi2 < 5.00000000000000024e-5

   1. Initial program 62.0%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-*.f64 46.7

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 46.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   6. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      9. lower-*.f64 46.8

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 46.8%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   8. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      2. unpow2 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      3. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      4. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      5. sqr-sin-a N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      8. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      10. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      11. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      12. mult-flip-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{\phi_1}{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{\phi_1}{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      14. mult-flip-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      15. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      16. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      17. lift-*.f64 45.6

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   9. Applied rewrites 45.6%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]
   10. Step-by-step derivation

       1. lift-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

       2. unpow2 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right)\right)}}\right) \]

       3. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right)\right)}}\right) \]

       4. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right)\right)}}\right) \]

       5. sqr-sin-a N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]

       8. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]

       9. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]

       10. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

       11. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

       12. mult-flip-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{\phi_1}{2}\right)\right)}}\right) \]

       13. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{\phi_1}{2}\right)\right)}}\right) \]

       14. mult-flip-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

       15. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

       16. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]

       17. lift-*.f64 45.6

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}\right) \]

   11. Applied rewrites 45.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}\right) \]

   if 5.00000000000000024e-5 < phi2

   1. Initial program 62.0%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 61.1%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 62.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 62.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}}\right) \]

   9. Applied rewrites 77.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right)}}\right) \]
   10. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       7. mult-flip-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       8. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       9. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       10. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       11. lift-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       12. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       13. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       14. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       15. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       17. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       18. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       19. mult-flip-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       20. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       21. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       22. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       23. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       24. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       25. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   11. Applied rewrites 78.2%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   12. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       7. mult-flip-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       8. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       9. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       10. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       11. lift-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       12. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       13. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       14. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       15. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       17. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       18. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       19. mult-flip-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       20. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       21. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       22. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       23. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       24. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       25. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   13. Applied rewrites 98.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   14. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}}}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   15. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \color{blue}{{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   16. Applied rewrites 57.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   17. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}}\right) \]
   18. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, \color{blue}{{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

   19. Applied rewrites 56.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)}}}\right) \]

   21. Applied rewrites 42.4%

       \[\leadsto \color{blue}{\left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right), \cos \phi_2, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\phi_2 \cdot 0.5\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right), \cos \phi_2, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\phi_2 \cdot 0.5\right)\right)\right)}} \cdot 2\right) \cdot R} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 29: 59.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} t_0 := 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\phi_2 \cdot 0.5\right)\right)\\ t_1 := 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\\ t_2 := \mathsf{fma}\left(t\_1, \cos \phi_2, t\_0\right)\\ t_3 := \mathsf{fma}\left(\cos \phi_1, t\_1, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)\\ t_4 := t\_0 + t\_1 \cdot \cos \phi_2\\ \mathbf{if}\;\phi_2 \leq -1.15 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_4}}{\sqrt{1 - t\_4}}\right)\\ \mathbf{elif}\;\phi_2 \leq 5 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_3}}{\sqrt{1 - t\_3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\tan^{-1}_* \frac{\sqrt{t\_2}}{\sqrt{1 - t\_2}} \cdot 2\right) \cdot R\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (- 0.5 (* 0.5 (cos (* 2.0 (* phi2 0.5))))))
        (t_1 (- 0.5 (* 0.5 (cos (* 2.0 (* (- lambda1 lambda2) 0.5))))))
        (t_2 (fma t_1 (cos phi2) t_0))
        (t_3 (fma (cos phi1) t_1 (pow (sin (* 0.5 phi1)) 2.0)))
        (t_4 (+ t_0 (* t_1 (cos phi2)))))
   (if (<= phi2 -1.15e-5)
     (* R (* 2.0 (atan2 (sqrt t_4) (sqrt (- 1.0 t_4)))))
     (if (<= phi2 5e-5)
       (* R (* 2.0 (atan2 (sqrt t_3) (sqrt (- 1.0 t_3)))))
       (* (* (atan2 (sqrt t_2) (sqrt (- 1.0 t_2))) 2.0) R)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = 0.5 - (0.5 * cos((2.0 * (phi2 * 0.5))));
	double t_1 = 0.5 - (0.5 * cos((2.0 * ((lambda1 - lambda2) * 0.5))));
	double t_2 = fma(t_1, cos(phi2), t_0);
	double t_3 = fma(cos(phi1), t_1, pow(sin((0.5 * phi1)), 2.0));
	double t_4 = t_0 + (t_1 * cos(phi2));
	double tmp;
	if (phi2 <= -1.15e-5) {
		tmp = R * (2.0 * atan2(sqrt(t_4), sqrt((1.0 - t_4))));
	} else if (phi2 <= 5e-5) {
		tmp = R * (2.0 * atan2(sqrt(t_3), sqrt((1.0 - t_3))));
	} else {
		tmp = (atan2(sqrt(t_2), sqrt((1.0 - t_2))) * 2.0) * R;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(phi2 * 0.5)))))
	t_1 = Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(Float64(lambda1 - lambda2) * 0.5)))))
	t_2 = fma(t_1, cos(phi2), t_0)
	t_3 = fma(cos(phi1), t_1, (sin(Float64(0.5 * phi1)) ^ 2.0))
	t_4 = Float64(t_0 + Float64(t_1 * cos(phi2)))
	tmp = 0.0
	if (phi2 <= -1.15e-5)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_4), sqrt(Float64(1.0 - t_4)))));
	elseif (phi2 <= 5e-5)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_3), sqrt(Float64(1.0 - t_3)))));
	else
		tmp = Float64(Float64(atan(sqrt(t_2), sqrt(Float64(1.0 - t_2))) * 2.0) * R);
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(phi2 * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(N[(lambda1 - lambda2), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[Cos[phi2], $MachinePrecision] + t$95$0), $MachinePrecision]}, Block[{t$95$3 = N[(N[Cos[phi1], $MachinePrecision] * t$95$1 + N[Power[N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$0 + N[(t$95$1 * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -1.15e-5], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$4], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$4), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 5e-5], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$3], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$3), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[ArcTan[N[Sqrt[t$95$2], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$2), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * R), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -1.15e-5

   1. Initial program 62.0%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 61.1%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 62.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 62.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}}\right) \]

   9. Applied rewrites 77.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right)}}\right) \]
   10. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       7. mult-flip-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       8. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       9. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       10. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       11. lift-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       12. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       13. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       14. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       15. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       17. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       18. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       19. mult-flip-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       20. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       21. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       22. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       23. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       24. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       25. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   11. Applied rewrites 78.2%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   12. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       7. mult-flip-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       8. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       9. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       10. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       11. lift-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       12. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       13. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       14. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       15. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       17. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       18. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       19. mult-flip-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       20. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       21. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       22. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       23. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       24. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       25. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   13. Applied rewrites 98.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   14. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}}}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   15. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \color{blue}{{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   16. Applied rewrites 57.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   17. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}}\right) \]
   18. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, \color{blue}{{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

   19. Applied rewrites 56.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)}}}\right) \]

   21. Applied rewrites 42.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\phi_2 \cdot 0.5\right)\right)\right) + \color{blue}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right) \cdot \cos \phi_2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)}}\right) \]

   23. Applied rewrites 42.4%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\phi_2 \cdot 0.5\right)\right)\right) + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right) \cdot \cos \phi_2}}{\sqrt{1 - \left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\phi_2 \cdot 0.5\right)\right)\right) + \color{blue}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right) \cdot \cos \phi_2}\right)}}\right) \]

   if -1.15e-5 < phi2 < 5.00000000000000024e-5

   1. Initial program 62.0%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-*.f64 46.7

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 46.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   6. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      9. lower-*.f64 46.8

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 46.8%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   8. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      3. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      4. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      5. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      7. pow2 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      8. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      9. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      10. sqr-sin-a N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot \frac{\lambda_1 - \lambda_2}{2}\right)}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      11. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot \frac{\lambda_1 - \lambda_2}{2}\right)}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot \frac{\lambda_1 - \lambda_2}{2}\right)}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      13. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{\lambda_1 - \lambda_2}{2}\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      14. lower-*.f64 44.4

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \frac{\lambda_1 - \lambda_2}{2}\right), {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

      15. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{\lambda_1 - \lambda_2}{2}\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      16. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      18. lower-*.f64 44.4

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right), {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   9. Applied rewrites 44.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, 0.5 - \color{blue}{0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]
   10. Step-by-step derivation

       1. lift-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

       2. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

       3. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

       4. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

       5. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

       6. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

       7. pow2 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

       8. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

       9. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

       10. sqr-sin-a N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot \frac{\lambda_1 - \lambda_2}{2}\right)}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

       11. lower--.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot \frac{\lambda_1 - \lambda_2}{2}\right)}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

       12. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot \frac{\lambda_1 - \lambda_2}{2}\right)}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

       13. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{\lambda_1 - \lambda_2}{2}\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

       14. lower-*.f64 44.4

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right), {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \frac{\lambda_1 - \lambda_2}{2}\right), {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

       15. lift-/.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{\lambda_1 - \lambda_2}{2}\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

       16. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

       17. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

       18. lower-*.f64 44.4

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right), {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right), {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   11. Applied rewrites 44.4%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right), {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, 0.5 - \color{blue}{0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   if 5.00000000000000024e-5 < phi2

   1. Initial program 62.0%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 61.1%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 62.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 62.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}}\right) \]

   9. Applied rewrites 77.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right)}}\right) \]
   10. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       7. mult-flip-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       8. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       9. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       10. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       11. lift-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       12. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       13. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       14. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       15. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       17. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       18. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       19. mult-flip-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       20. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       21. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       22. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       23. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       24. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       25. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   11. Applied rewrites 78.2%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   12. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       7. mult-flip-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       8. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       9. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       10. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       11. lift-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       12. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       13. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       14. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       15. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       17. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       18. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       19. mult-flip-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       20. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       21. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       22. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       23. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       24. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       25. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   13. Applied rewrites 98.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   14. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}}}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   15. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \color{blue}{{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   16. Applied rewrites 57.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   17. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}}\right) \]
   18. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, \color{blue}{{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

   19. Applied rewrites 56.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)}}}\right) \]

   21. Applied rewrites 42.4%

       \[\leadsto \color{blue}{\left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right), \cos \phi_2, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\phi_2 \cdot 0.5\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right), \cos \phi_2, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\phi_2 \cdot 0.5\right)\right)\right)}} \cdot 2\right) \cdot R} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 30: 57.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} t_0 := 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\\ t_1 := 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\\ t_2 := 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\\ t_3 := \mathsf{fma}\left(t\_1, \cos \phi_1, t\_2\right)\\ t_4 := t\_1 \cdot \cos \phi_2\\ t_5 := t\_2 + t\_1 \cdot \cos \phi_1\\ \mathbf{if}\;\phi_1 \leq -0.42:\\ \;\;\;\;\left(R \cdot \tan^{-1}_* \frac{\sqrt{t\_3}}{\sqrt{1 - t\_3}}\right) \cdot 2\\ \mathbf{elif}\;\phi_1 \leq 7.3:\\ \;\;\;\;\left(\tan^{-1}_* \frac{\sqrt{t\_4 + t\_0}}{\sqrt{\left(1 - t\_0\right) - t\_4}} \cdot 2\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_5}}{\sqrt{1 - t\_5}}\right)\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (- 0.5 (* 0.5 (cos (* 2.0 (* (- phi1 phi2) 0.5))))))
        (t_1 (- 0.5 (* 0.5 (cos (* 2.0 (* (- lambda1 lambda2) 0.5))))))
        (t_2 (- 0.5 (* 0.5 (cos (* 2.0 (* 0.5 phi1))))))
        (t_3 (fma t_1 (cos phi1) t_2))
        (t_4 (* t_1 (cos phi2)))
        (t_5 (+ t_2 (* t_1 (cos phi1)))))
   (if (<= phi1 -0.42)
     (* (* R (atan2 (sqrt t_3) (sqrt (- 1.0 t_3)))) 2.0)
     (if (<= phi1 7.3)
       (* (* (atan2 (sqrt (+ t_4 t_0)) (sqrt (- (- 1.0 t_0) t_4))) 2.0) R)
       (* R (* 2.0 (atan2 (sqrt t_5) (sqrt (- 1.0 t_5)))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = 0.5 - (0.5 * cos((2.0 * ((phi1 - phi2) * 0.5))));
	double t_1 = 0.5 - (0.5 * cos((2.0 * ((lambda1 - lambda2) * 0.5))));
	double t_2 = 0.5 - (0.5 * cos((2.0 * (0.5 * phi1))));
	double t_3 = fma(t_1, cos(phi1), t_2);
	double t_4 = t_1 * cos(phi2);
	double t_5 = t_2 + (t_1 * cos(phi1));
	double tmp;
	if (phi1 <= -0.42) {
		tmp = (R * atan2(sqrt(t_3), sqrt((1.0 - t_3)))) * 2.0;
	} else if (phi1 <= 7.3) {
		tmp = (atan2(sqrt((t_4 + t_0)), sqrt(((1.0 - t_0) - t_4))) * 2.0) * R;
	} else {
		tmp = R * (2.0 * atan2(sqrt(t_5), sqrt((1.0 - t_5))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(Float64(phi1 - phi2) * 0.5)))))
	t_1 = Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(Float64(lambda1 - lambda2) * 0.5)))))
	t_2 = Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(0.5 * phi1)))))
	t_3 = fma(t_1, cos(phi1), t_2)
	t_4 = Float64(t_1 * cos(phi2))
	t_5 = Float64(t_2 + Float64(t_1 * cos(phi1)))
	tmp = 0.0
	if (phi1 <= -0.42)
		tmp = Float64(Float64(R * atan(sqrt(t_3), sqrt(Float64(1.0 - t_3)))) * 2.0);
	elseif (phi1 <= 7.3)
		tmp = Float64(Float64(atan(sqrt(Float64(t_4 + t_0)), sqrt(Float64(Float64(1.0 - t_0) - t_4))) * 2.0) * R);
	else
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_5), sqrt(Float64(1.0 - t_5)))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(N[(phi1 - phi2), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(N[(lambda1 - lambda2), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(0.5 * phi1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 * N[Cos[phi1], $MachinePrecision] + t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$1 * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$2 + N[(t$95$1 * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi1, -0.42], N[(N[(R * N[ArcTan[N[Sqrt[t$95$3], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$3), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], If[LessEqual[phi1, 7.3], N[(N[(N[ArcTan[N[Sqrt[N[(t$95$4 + t$95$0), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[(1.0 - t$95$0), $MachinePrecision] - t$95$4), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * R), $MachinePrecision], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$5], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -0.419999999999999984

   1. Initial program 62.0%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-*.f64 46.7

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 46.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   6. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      9. lower-*.f64 46.8

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 46.8%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]

   9. Applied rewrites 43.1%

      \[\leadsto \color{blue}{\left(R \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}\right) \cdot 2} \]

   if -0.419999999999999984 < phi1 < 7.29999999999999982

   1. Initial program 62.0%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 52.8

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 52.8%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_2 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}\right)}}\right) \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}\right) \]

      6. lower--.f64 50.7

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 50.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_2 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}\right)}}\right) \]

   9. Applied rewrites 45.7%

      \[\leadsto \color{blue}{\left(\tan^{-1}_* \frac{\sqrt{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right) \cdot \cos \phi_2 + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)}}{\sqrt{\left(1 - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)\right) - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right) \cdot \cos \phi_2}} \cdot 2\right) \cdot R} \]

   if 7.29999999999999982 < phi1

   1. Initial program 62.0%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-*.f64 46.7

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 46.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   6. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      9. lower-*.f64 46.8

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 46.8%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   8. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + \color{blue}{{\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      2. +-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      3. lower-+.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   9. Applied rewrites 43.0%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right) + \color{blue}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right) \cdot \cos \phi_1}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]
   10. Step-by-step derivation

       1. lift-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right) + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \cos \phi_1}}{\sqrt{1 - \left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + \color{blue}{{\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}\right)}}\right) \]

       2. +-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right) + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \cos \phi_1}}{\sqrt{1 - \left({\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}\right)}}\right) \]

       3. lower-+.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right) + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \cos \phi_1}}{\sqrt{1 - \left({\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}\right)}}\right) \]

   11. Applied rewrites 43.1%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right) + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right) \cdot \cos \phi_1}}{\sqrt{1 - \left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right) + \color{blue}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right) \cdot \cos \phi_1}\right)}}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 31: 57.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} t_0 := 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\\ t_1 := 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\\ t_2 := t\_0 + t\_1 \cdot \cos \phi_1\\ t_3 := \mathsf{fma}\left(t\_1, \cos \phi_2, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\phi_2 \cdot 0.5\right)\right)\right)\\ t_4 := \mathsf{fma}\left(t\_1, \cos \phi_1, t\_0\right)\\ \mathbf{if}\;\phi_1 \leq -4.5:\\ \;\;\;\;\left(R \cdot \tan^{-1}_* \frac{\sqrt{t\_4}}{\sqrt{1 - t\_4}}\right) \cdot 2\\ \mathbf{elif}\;\phi_1 \leq 1900:\\ \;\;\;\;\left(\tan^{-1}_* \frac{\sqrt{t\_3}}{\sqrt{1 - t\_3}} \cdot 2\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_2}}{\sqrt{1 - t\_2}}\right)\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (- 0.5 (* 0.5 (cos (* 2.0 (* 0.5 phi1))))))
        (t_1 (- 0.5 (* 0.5 (cos (* 2.0 (* (- lambda1 lambda2) 0.5))))))
        (t_2 (+ t_0 (* t_1 (cos phi1))))
        (t_3 (fma t_1 (cos phi2) (- 0.5 (* 0.5 (cos (* 2.0 (* phi2 0.5)))))))
        (t_4 (fma t_1 (cos phi1) t_0)))
   (if (<= phi1 -4.5)
     (* (* R (atan2 (sqrt t_4) (sqrt (- 1.0 t_4)))) 2.0)
     (if (<= phi1 1900.0)
       (* (* (atan2 (sqrt t_3) (sqrt (- 1.0 t_3))) 2.0) R)
       (* R (* 2.0 (atan2 (sqrt t_2) (sqrt (- 1.0 t_2)))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = 0.5 - (0.5 * cos((2.0 * (0.5 * phi1))));
	double t_1 = 0.5 - (0.5 * cos((2.0 * ((lambda1 - lambda2) * 0.5))));
	double t_2 = t_0 + (t_1 * cos(phi1));
	double t_3 = fma(t_1, cos(phi2), (0.5 - (0.5 * cos((2.0 * (phi2 * 0.5))))));
	double t_4 = fma(t_1, cos(phi1), t_0);
	double tmp;
	if (phi1 <= -4.5) {
		tmp = (R * atan2(sqrt(t_4), sqrt((1.0 - t_4)))) * 2.0;
	} else if (phi1 <= 1900.0) {
		tmp = (atan2(sqrt(t_3), sqrt((1.0 - t_3))) * 2.0) * R;
	} else {
		tmp = R * (2.0 * atan2(sqrt(t_2), sqrt((1.0 - t_2))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(0.5 * phi1)))))
	t_1 = Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(Float64(lambda1 - lambda2) * 0.5)))))
	t_2 = Float64(t_0 + Float64(t_1 * cos(phi1)))
	t_3 = fma(t_1, cos(phi2), Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(phi2 * 0.5))))))
	t_4 = fma(t_1, cos(phi1), t_0)
	tmp = 0.0
	if (phi1 <= -4.5)
		tmp = Float64(Float64(R * atan(sqrt(t_4), sqrt(Float64(1.0 - t_4)))) * 2.0);
	elseif (phi1 <= 1900.0)
		tmp = Float64(Float64(atan(sqrt(t_3), sqrt(Float64(1.0 - t_3))) * 2.0) * R);
	else
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_2), sqrt(Float64(1.0 - t_2)))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(0.5 * phi1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(N[(lambda1 - lambda2), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 + N[(t$95$1 * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 * N[Cos[phi2], $MachinePrecision] + N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(phi2 * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$1 * N[Cos[phi1], $MachinePrecision] + t$95$0), $MachinePrecision]}, If[LessEqual[phi1, -4.5], N[(N[(R * N[ArcTan[N[Sqrt[t$95$4], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$4), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], If[LessEqual[phi1, 1900.0], N[(N[(N[ArcTan[N[Sqrt[t$95$3], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$3), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * R), $MachinePrecision], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$2], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$2), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -4.5

   1. Initial program 62.0%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-*.f64 46.7

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 46.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   6. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      9. lower-*.f64 46.8

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 46.8%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]

   9. Applied rewrites 43.1%

      \[\leadsto \color{blue}{\left(R \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}\right) \cdot 2} \]

   if -4.5 < phi1 < 1900

   1. Initial program 62.0%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 61.1%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 62.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 62.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}}\right) \]

   9. Applied rewrites 77.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right)}}\right) \]
   10. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       7. mult-flip-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       8. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       9. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       10. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       11. lift-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       12. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       13. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       14. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       15. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       17. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       18. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       19. mult-flip-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       20. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       21. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       22. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       23. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       24. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       25. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   11. Applied rewrites 78.2%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   12. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       7. mult-flip-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       8. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       9. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       10. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       11. lift-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       12. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       13. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       14. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       15. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       17. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       18. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       19. mult-flip-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       20. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       21. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       22. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       23. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       24. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       25. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   13. Applied rewrites 98.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   14. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}}}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   15. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \color{blue}{{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   16. Applied rewrites 57.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   17. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}}\right) \]
   18. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, \color{blue}{{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

   19. Applied rewrites 56.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)}}}\right) \]

   21. Applied rewrites 42.4%

       \[\leadsto \color{blue}{\left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right), \cos \phi_2, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\phi_2 \cdot 0.5\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right), \cos \phi_2, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\phi_2 \cdot 0.5\right)\right)\right)}} \cdot 2\right) \cdot R} \]

   if 1900 < phi1

   1. Initial program 62.0%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-*.f64 46.7

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 46.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   6. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      9. lower-*.f64 46.8

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 46.8%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   8. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + \color{blue}{{\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      2. +-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      3. lower-+.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   9. Applied rewrites 43.0%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right) + \color{blue}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right) \cdot \cos \phi_1}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]
   10. Step-by-step derivation

       1. lift-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right) + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \cos \phi_1}}{\sqrt{1 - \left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + \color{blue}{{\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}\right)}}\right) \]

       2. +-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right) + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \cos \phi_1}}{\sqrt{1 - \left({\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}\right)}}\right) \]

       3. lower-+.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right) + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \cos \phi_1}}{\sqrt{1 - \left({\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}\right)}}\right) \]

   11. Applied rewrites 43.1%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right) + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right) \cdot \cos \phi_1}}{\sqrt{1 - \left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right) + \color{blue}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right) \cdot \cos \phi_1}\right)}}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 32: 57.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} t_0 := 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\\ t_1 := \mathsf{fma}\left(t\_0, \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)\\ t_2 := \left(R \cdot \tan^{-1}_* \frac{\sqrt{t\_1}}{\sqrt{1 - t\_1}}\right) \cdot 2\\ t_3 := \mathsf{fma}\left(t\_0, \cos \phi_2, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\phi_2 \cdot 0.5\right)\right)\right)\\ \mathbf{if}\;\phi_1 \leq -4.5:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\phi_1 \leq 1900:\\ \;\;\;\;\left(\tan^{-1}_* \frac{\sqrt{t\_3}}{\sqrt{1 - t\_3}} \cdot 2\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (- 0.5 (* 0.5 (cos (* 2.0 (* (- lambda1 lambda2) 0.5))))))
        (t_1 (fma t_0 (cos phi1) (- 0.5 (* 0.5 (cos (* 2.0 (* 0.5 phi1)))))))
        (t_2 (* (* R (atan2 (sqrt t_1) (sqrt (- 1.0 t_1)))) 2.0))
        (t_3 (fma t_0 (cos phi2) (- 0.5 (* 0.5 (cos (* 2.0 (* phi2 0.5))))))))
   (if (<= phi1 -4.5)
     t_2
     (if (<= phi1 1900.0)
       (* (* (atan2 (sqrt t_3) (sqrt (- 1.0 t_3))) 2.0) R)
       t_2))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = 0.5 - (0.5 * cos((2.0 * ((lambda1 - lambda2) * 0.5))));
	double t_1 = fma(t_0, cos(phi1), (0.5 - (0.5 * cos((2.0 * (0.5 * phi1))))));
	double t_2 = (R * atan2(sqrt(t_1), sqrt((1.0 - t_1)))) * 2.0;
	double t_3 = fma(t_0, cos(phi2), (0.5 - (0.5 * cos((2.0 * (phi2 * 0.5))))));
	double tmp;
	if (phi1 <= -4.5) {
		tmp = t_2;
	} else if (phi1 <= 1900.0) {
		tmp = (atan2(sqrt(t_3), sqrt((1.0 - t_3))) * 2.0) * R;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(Float64(lambda1 - lambda2) * 0.5)))))
	t_1 = fma(t_0, cos(phi1), Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(0.5 * phi1))))))
	t_2 = Float64(Float64(R * atan(sqrt(t_1), sqrt(Float64(1.0 - t_1)))) * 2.0)
	t_3 = fma(t_0, cos(phi2), Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(phi2 * 0.5))))))
	tmp = 0.0
	if (phi1 <= -4.5)
		tmp = t_2;
	elseif (phi1 <= 1900.0)
		tmp = Float64(Float64(atan(sqrt(t_3), sqrt(Float64(1.0 - t_3))) * 2.0) * R);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(N[(lambda1 - lambda2), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[Cos[phi1], $MachinePrecision] + N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(0.5 * phi1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(R * N[ArcTan[N[Sqrt[t$95$1], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$0 * N[Cos[phi2], $MachinePrecision] + N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(phi2 * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi1, -4.5], t$95$2, If[LessEqual[phi1, 1900.0], N[(N[(N[ArcTan[N[Sqrt[t$95$3], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$3), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * R), $MachinePrecision], t$95$2]]]]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -4.5 or 1900 < phi1

   1. Initial program 62.0%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-*.f64 46.7

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 46.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   6. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      9. lower-*.f64 46.8

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 46.8%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]

   9. Applied rewrites 43.1%

      \[\leadsto \color{blue}{\left(R \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}\right) \cdot 2} \]

   if -4.5 < phi1 < 1900

   1. Initial program 62.0%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 61.1%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 62.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 62.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}}\right) \]

   9. Applied rewrites 77.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right)}}\right) \]
   10. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       7. mult-flip-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       8. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       9. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       10. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       11. lift-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       12. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       13. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       14. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       15. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       17. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       18. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       19. mult-flip-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       20. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       21. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       22. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       23. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       24. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       25. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   11. Applied rewrites 78.2%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   12. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       7. mult-flip-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       8. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       9. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       10. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       11. lift-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       12. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       13. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       14. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       15. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       17. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       18. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       19. mult-flip-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       20. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       21. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       22. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       23. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       24. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       25. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   13. Applied rewrites 98.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   14. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}}}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   15. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \color{blue}{{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   16. Applied rewrites 57.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   17. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}}\right) \]
   18. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, \color{blue}{{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

   19. Applied rewrites 56.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)}}}\right) \]

   21. Applied rewrites 42.4%

       \[\leadsto \color{blue}{\left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right), \cos \phi_2, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\phi_2 \cdot 0.5\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right), \cos \phi_2, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\phi_2 \cdot 0.5\right)\right)\right)}} \cdot 2\right) \cdot R} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 33: 50.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)\\ t_1 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_2 := {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\\ t_3 := \left(R \cdot \tan^{-1}_* \frac{\sqrt{t\_0}}{\sqrt{1 - t\_0}}\right) \cdot 2\\ \mathbf{if}\;t\_1 \leq -0.001:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-9}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_2}}{\sqrt{1 - \mathsf{fma}\left(0.5, {\sin \left(-0.5 \cdot \lambda_2\right)}^{2} \cdot \left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_2 - \phi_1\right)\right), t\_2\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (fma
          (- 0.5 (* 0.5 (cos (* 2.0 (* (- lambda1 lambda2) 0.5)))))
          (cos phi1)
          (- 0.5 (* 0.5 (cos (* 2.0 (* 0.5 phi1)))))))
        (t_1 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_2 (pow (sin (* 0.5 (- phi1 phi2))) 2.0))
        (t_3 (* (* R (atan2 (sqrt t_0) (sqrt (- 1.0 t_0)))) 2.0)))
   (if (<= t_1 -0.001)
     t_3
     (if (<= t_1 5e-9)
       (*
        R
        (*
         2.0
         (atan2
          (sqrt t_2)
          (sqrt
           (-
            1.0
            (fma
             0.5
             (*
              (pow (sin (* -0.5 lambda2)) 2.0)
              (+ (cos (+ phi1 phi2)) (cos (- phi2 phi1))))
             t_2))))))
       t_3))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = fma((0.5 - (0.5 * cos((2.0 * ((lambda1 - lambda2) * 0.5))))), cos(phi1), (0.5 - (0.5 * cos((2.0 * (0.5 * phi1))))));
	double t_1 = sin(((lambda1 - lambda2) / 2.0));
	double t_2 = pow(sin((0.5 * (phi1 - phi2))), 2.0);
	double t_3 = (R * atan2(sqrt(t_0), sqrt((1.0 - t_0)))) * 2.0;
	double tmp;
	if (t_1 <= -0.001) {
		tmp = t_3;
	} else if (t_1 <= 5e-9) {
		tmp = R * (2.0 * atan2(sqrt(t_2), sqrt((1.0 - fma(0.5, (pow(sin((-0.5 * lambda2)), 2.0) * (cos((phi1 + phi2)) + cos((phi2 - phi1)))), t_2)))));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = fma(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(Float64(lambda1 - lambda2) * 0.5))))), cos(phi1), Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(0.5 * phi1))))))
	t_1 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_2 = sin(Float64(0.5 * Float64(phi1 - phi2))) ^ 2.0
	t_3 = Float64(Float64(R * atan(sqrt(t_0), sqrt(Float64(1.0 - t_0)))) * 2.0)
	tmp = 0.0
	if (t_1 <= -0.001)
		tmp = t_3;
	elseif (t_1 <= 5e-9)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_2), sqrt(Float64(1.0 - fma(0.5, Float64((sin(Float64(-0.5 * lambda2)) ^ 2.0) * Float64(cos(Float64(phi1 + phi2)) + cos(Float64(phi2 - phi1)))), t_2))))));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(N[(lambda1 - lambda2), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision] + N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(0.5 * phi1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[N[(0.5 * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(N[(R * N[ArcTan[N[Sqrt[t$95$0], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]}, If[LessEqual[t$95$1, -0.001], t$95$3, If[LessEqual[t$95$1, 5e-9], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$2], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(0.5 * N[(N[Power[N[Sin[N[(-0.5 * lambda2), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Cos[N[(phi1 + phi2), $MachinePrecision]], $MachinePrecision] + N[Cos[N[(phi2 - phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64))) < -1e-3 or 5.0000000000000001e-9 < (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))

   1. Initial program 62.0%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-*.f64 46.7

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 46.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   6. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      9. lower-*.f64 46.8

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 46.8%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]

   9. Applied rewrites 43.1%

      \[\leadsto \color{blue}{\left(R \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}\right) \cdot 2} \]

   if -1e-3 < (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64))) < 5.0000000000000001e-9

   1. Initial program 62.0%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in lambda1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\color{blue}{\sqrt{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}\right) + {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. lower-sqrt.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}\right) + {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower--.f64 45.6

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 45.6%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\color{blue}{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right)} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\color{blue}{\cos \phi_1} \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\cos \phi_2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. cos-mult N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\color{blue}{\frac{\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_1 - \phi_2\right)}{2}} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\color{blue}{\frac{1}{\frac{2}{\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_1 - \phi_2\right)}}} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower-unsound-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\color{blue}{\frac{1}{\frac{2}{\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_1 - \phi_2\right)}}} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-unsound-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\frac{1}{\color{blue}{\frac{2}{\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_1 - \phi_2\right)}}} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. +-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\frac{1}{\frac{2}{\color{blue}{\cos \left(\phi_1 - \phi_2\right) + \cos \left(\phi_1 + \phi_2\right)}}} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-+.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\frac{1}{\frac{2}{\color{blue}{\cos \left(\phi_1 - \phi_2\right) + \cos \left(\phi_1 + \phi_2\right)}}} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\frac{1}{\frac{2}{\cos \color{blue}{\left(\phi_1 - \phi_2\right)} + \cos \left(\phi_1 + \phi_2\right)}} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. cos-neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\frac{1}{\frac{2}{\color{blue}{\cos \left(\mathsf{neg}\left(\left(\phi_1 - \phi_2\right)\right)\right)} + \cos \left(\phi_1 + \phi_2\right)}} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\frac{1}{\frac{2}{\color{blue}{\cos \left(\mathsf{neg}\left(\left(\phi_1 - \phi_2\right)\right)\right)} + \cos \left(\phi_1 + \phi_2\right)}} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\frac{1}{\frac{2}{\cos \left(\mathsf{neg}\left(\color{blue}{\left(\phi_1 - \phi_2\right)}\right)\right) + \cos \left(\phi_1 + \phi_2\right)}} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. sub-negate-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\frac{1}{\frac{2}{\cos \color{blue}{\left(\phi_2 - \phi_1\right)} + \cos \left(\phi_1 + \phi_2\right)}} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\frac{1}{\frac{2}{\cos \color{blue}{\left(\phi_2 - \phi_1\right)} + \cos \left(\phi_1 + \phi_2\right)}} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\frac{1}{\frac{2}{\cos \left(\phi_2 - \phi_1\right) + \color{blue}{\cos \left(\phi_1 + \phi_2\right)}}} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. +-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\frac{1}{\frac{2}{\cos \left(\phi_2 - \phi_1\right) + \cos \color{blue}{\left(\phi_2 + \phi_1\right)}}} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-+.f64 46.1

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\frac{1}{\frac{2}{\cos \left(\phi_2 - \phi_1\right) + \cos \color{blue}{\left(\phi_2 + \phi_1\right)}}} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   6. Applied rewrites 46.1%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\color{blue}{\frac{1}{\frac{2}{\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)}}} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Taylor expanded in lambda1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\frac{1}{2} \cdot \left({\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2} \cdot \left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_2 - \phi_1\right)\right)\right) + {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}}\right) \]
   8. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2}, \color{blue}{{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2} \cdot \left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_2 - \phi_1\right)\right)}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2}, {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2} \cdot \color{blue}{\left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_2 - \phi_1\right)\right)}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2}, {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2} \cdot \left(\color{blue}{\cos \left(\phi_1 + \phi_2\right)} + \cos \left(\phi_2 - \phi_1\right)\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2}, {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2} \cdot \left(\cos \color{blue}{\left(\phi_1 + \phi_2\right)} + \cos \left(\phi_2 - \phi_1\right)\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2}, {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2} \cdot \left(\cos \left(\color{blue}{\phi_1} + \phi_2\right) + \cos \left(\phi_2 - \phi_1\right)\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

      6. lower-+.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2}, {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2} \cdot \left(\cos \left(\phi_1 + \phi_2\right) + \color{blue}{\cos \left(\phi_2 - \phi_1\right)}\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

      7. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2}, {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2} \cdot \left(\cos \left(\phi_1 + \phi_2\right) + \cos \color{blue}{\left(\phi_2 - \phi_1\right)}\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

      8. lower-+.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2}, {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2} \cdot \left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\color{blue}{\phi_2} - \phi_1\right)\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

      9. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2}, {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2} \cdot \left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_2 - \phi_1\right)\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

      10. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2}, {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2} \cdot \left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_2 - \phi_1\right)\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

   9. Applied rewrites 45.8%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(0.5, {\sin \left(-0.5 \cdot \lambda_2\right)}^{2} \cdot \left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_2 - \phi_1\right)\right), {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}}\right) \]

   10. Taylor expanded in lambda2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2}, {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2} \cdot \left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_2 - \phi_1\right)\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]
   11. Step-by-step derivation

       1. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2}, {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2} \cdot \left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_2 - \phi_1\right)\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       2. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2}, {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2} \cdot \left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_2 - \phi_1\right)\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2}, {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2} \cdot \left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_2 - \phi_1\right)\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       4. lower--.f64 29.5

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(0.5, {\sin \left(-0.5 \cdot \lambda_2\right)}^{2} \cdot \left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_2 - \phi_1\right)\right), {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

   12. Applied rewrites 29.5%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(0.5, {\sin \left(-0.5 \cdot \lambda_2\right)}^{2} \cdot \left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_2 - \phi_1\right)\right), {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 34: 43.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(\cos \phi_1, 0.5 - 0.5 \cdot \cos \lambda_1, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)\\ t_1 := \left(\tan^{-1}_* \frac{\sqrt{t\_0}}{\sqrt{1 - t\_0}} \cdot 2\right) \cdot R\\ \mathbf{if}\;\phi_1 \leq -3.3 \cdot 10^{-10}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\phi_1 \leq 7 \cdot 10^{-40}:\\ \;\;\;\;\left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot 0.25\right)}} \cdot 2\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (fma
          (cos phi1)
          (- 0.5 (* 0.5 (cos lambda1)))
          (pow (sin (* 0.5 phi1)) 2.0)))
        (t_1 (* (* (atan2 (sqrt t_0) (sqrt (- 1.0 t_0))) 2.0) R)))
   (if (<= phi1 -3.3e-10)
     t_1
     (if (<= phi1 7e-40)
       (*
        (*
         (atan2
          (sqrt
           (fma
            (cos phi2)
            (pow (sin (* 0.5 (- lambda1 lambda2))) 2.0)
            (pow (sin (* -0.5 phi2)) 2.0)))
          (sqrt
           (-
            1.0
            (fma
             (- 0.5 (* (cos (* 1.0 (- lambda1 lambda2))) 0.5))
             (cos phi1)
             (* (* phi1 phi1) 0.25)))))
         2.0)
        R)
       t_1))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = fma(cos(phi1), (0.5 - (0.5 * cos(lambda1))), pow(sin((0.5 * phi1)), 2.0));
	double t_1 = (atan2(sqrt(t_0), sqrt((1.0 - t_0))) * 2.0) * R;
	double tmp;
	if (phi1 <= -3.3e-10) {
		tmp = t_1;
	} else if (phi1 <= 7e-40) {
		tmp = (atan2(sqrt(fma(cos(phi2), pow(sin((0.5 * (lambda1 - lambda2))), 2.0), pow(sin((-0.5 * phi2)), 2.0))), sqrt((1.0 - fma((0.5 - (cos((1.0 * (lambda1 - lambda2))) * 0.5)), cos(phi1), ((phi1 * phi1) * 0.25))))) * 2.0) * R;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = fma(cos(phi1), Float64(0.5 - Float64(0.5 * cos(lambda1))), (sin(Float64(0.5 * phi1)) ^ 2.0))
	t_1 = Float64(Float64(atan(sqrt(t_0), sqrt(Float64(1.0 - t_0))) * 2.0) * R)
	tmp = 0.0
	if (phi1 <= -3.3e-10)
		tmp = t_1;
	elseif (phi1 <= 7e-40)
		tmp = Float64(Float64(atan(sqrt(fma(cos(phi2), (sin(Float64(0.5 * Float64(lambda1 - lambda2))) ^ 2.0), (sin(Float64(-0.5 * phi2)) ^ 2.0))), sqrt(Float64(1.0 - fma(Float64(0.5 - Float64(cos(Float64(1.0 * Float64(lambda1 - lambda2))) * 0.5)), cos(phi1), Float64(Float64(phi1 * phi1) * 0.25))))) * 2.0) * R);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi1], $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[ArcTan[N[Sqrt[t$95$0], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * R), $MachinePrecision]}, If[LessEqual[phi1, -3.3e-10], t$95$1, If[LessEqual[phi1, 7e-40], N[(N[(N[ArcTan[N[Sqrt[N[(N[Cos[phi2], $MachinePrecision] * N[Power[N[Sin[N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(N[(0.5 - N[(N[Cos[N[(1.0 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision] + N[(N[(phi1 * phi1), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * R), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -3.3e-10 or 7.0000000000000003e-40 < phi1

   1. Initial program 62.0%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-*.f64 46.7

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 46.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   6. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      9. lower-*.f64 46.8

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 46.8%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]

   8. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      2. lower-pow.f64 31.8

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   10. Applied rewrites 31.8%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   11. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}\right) \]
   12. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}\right) \]

       2. lower-pow.f64 22.3

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}\right) \]

   13. Applied rewrites 22.3%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}\right) \]

   15. Applied rewrites 19.9%

       \[\leadsto \color{blue}{\left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot 0.25\right)}}{\sqrt{1 - \mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot 0.25\right)}} \cdot 2\right) \cdot R} \]

   16. Taylor expanded in lambda2 around 0

       \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_1\right) + \color{blue}{{\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}} \cdot 2\right) \cdot R \]
   17. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \lambda_1}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}} \cdot 2\right) \cdot R \]

       2. lower-cos.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \color{blue}{\frac{1}{2}} \cdot \cos \lambda_1, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}} \cdot 2\right) \cdot R \]

       3. lower--.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \lambda_1}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}} \cdot 2\right) \cdot R \]

       4. lower-*.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_1, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}} \cdot 2\right) \cdot R \]

       5. lower-cos.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_1, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}} \cdot 2\right) \cdot R \]

       6. lower-pow.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_1, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}} \cdot 2\right) \cdot R \]

       7. lower-sin.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_1, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}} \cdot 2\right) \cdot R \]

       8. lower-*.f64 13.2

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, 0.5 - 0.5 \cdot \cos \lambda_1, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot 0.25\right)}} \cdot 2\right) \cdot R \]

   18. Applied rewrites 13.2%

       \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{0.5 - 0.5 \cdot \cos \lambda_1}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot 0.25\right)}} \cdot 2\right) \cdot R \]

   19. Taylor expanded in lambda2 around 0

       \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_1, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left(\cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_1\right) + \color{blue}{{\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}\right)}} \cdot 2\right) \cdot R \]
   20. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_1, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \lambda_1}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}} \cdot 2\right) \cdot R \]

       2. lower-cos.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_1, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \color{blue}{\frac{1}{2}} \cdot \cos \lambda_1, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}} \cdot 2\right) \cdot R \]

       3. lower--.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_1, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \lambda_1}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}} \cdot 2\right) \cdot R \]

       4. lower-*.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_1, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_1, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}} \cdot 2\right) \cdot R \]

       5. lower-cos.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_1, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_1, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}} \cdot 2\right) \cdot R \]

       6. lower-pow.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_1, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_1, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}} \cdot 2\right) \cdot R \]

       7. lower-sin.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_1, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_1, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}} \cdot 2\right) \cdot R \]

       8. lower-*.f64 33.2

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, 0.5 - 0.5 \cdot \cos \lambda_1, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, 0.5 - 0.5 \cdot \cos \lambda_1, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}} \cdot 2\right) \cdot R \]

   21. Applied rewrites 33.2%

       \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, 0.5 - 0.5 \cdot \cos \lambda_1, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{0.5 - 0.5 \cdot \cos \lambda_1}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}} \cdot 2\right) \cdot R \]

   if -3.3e-10 < phi1 < 7.0000000000000003e-40

   1. Initial program 62.0%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-*.f64 46.7

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 46.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   6. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      9. lower-*.f64 46.8

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 46.8%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]

   8. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      2. lower-pow.f64 31.8

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   10. Applied rewrites 31.8%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   11. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}\right) \]
   12. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}\right) \]

       2. lower-pow.f64 22.3

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}\right) \]

   13. Applied rewrites 22.3%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}\right) \]

   15. Applied rewrites 19.9%

       \[\leadsto \color{blue}{\left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot 0.25\right)}}{\sqrt{1 - \mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot 0.25\right)}} \cdot 2\right) \cdot R} \]

   16. Taylor expanded in phi1 around 0

       \[\leadsto \left(\tan^{-1}_* \frac{\color{blue}{\sqrt{\cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}}}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}} \cdot 2\right) \cdot R \]
   17. Step-by-step derivation

       1. lower-sqrt.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}} \cdot 2\right) \cdot R \]

       2. lower-fma.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}} \cdot 2\right) \cdot R \]

       3. lower-cos.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}} \cdot 2\right) \cdot R \]

       4. lower-pow.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}} \cdot 2\right) \cdot R \]

       5. lower-sin.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}} \cdot 2\right) \cdot R \]

       6. lower-*.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}} \cdot 2\right) \cdot R \]

       7. lower--.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}} \cdot 2\right) \cdot R \]

       8. lower-pow.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}} \cdot 2\right) \cdot R \]

       9. lower-sin.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}} \cdot 2\right) \cdot R \]

       10. lower-*.f64 23.6

           \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot 0.25\right)}} \cdot 2\right) \cdot R \]

   18. Applied rewrites 23.6%

       \[\leadsto \left(\tan^{-1}_* \frac{\color{blue}{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}}{\sqrt{1 - \mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot 0.25\right)}} \cdot 2\right) \cdot R \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 35: 40.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_0 := {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\\ t_1 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_2 := R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, t\_0, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, t\_0, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}\right)\\ t_3 := {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-10}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 10^{-42}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_3}}{\sqrt{1 - \mathsf{fma}\left(0.5, {\sin \left(-0.5 \cdot \lambda_2\right)}^{2} \cdot \left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_2 - \phi_1\right)\right), t\_3\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (pow (sin (* 0.5 (- lambda1 lambda2))) 2.0))
        (t_1 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_2
         (*
          R
          (*
           2.0
           (atan2
            (sqrt (fma (cos phi1) t_0 (* 0.25 (pow phi1 2.0))))
            (sqrt
             (- 1.0 (fma (cos phi2) t_0 (pow (sin (* -0.5 phi2)) 2.0))))))))
        (t_3 (pow (sin (* 0.5 (- phi1 phi2))) 2.0)))
   (if (<= t_1 -1e-10)
     t_2
     (if (<= t_1 1e-42)
       (*
        R
        (*
         2.0
         (atan2
          (sqrt t_3)
          (sqrt
           (-
            1.0
            (fma
             0.5
             (*
              (pow (sin (* -0.5 lambda2)) 2.0)
              (+ (cos (+ phi1 phi2)) (cos (- phi2 phi1))))
             t_3))))))
       t_2))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = pow(sin((0.5 * (lambda1 - lambda2))), 2.0);
	double t_1 = sin(((lambda1 - lambda2) / 2.0));
	double t_2 = R * (2.0 * atan2(sqrt(fma(cos(phi1), t_0, (0.25 * pow(phi1, 2.0)))), sqrt((1.0 - fma(cos(phi2), t_0, pow(sin((-0.5 * phi2)), 2.0))))));
	double t_3 = pow(sin((0.5 * (phi1 - phi2))), 2.0);
	double tmp;
	if (t_1 <= -1e-10) {
		tmp = t_2;
	} else if (t_1 <= 1e-42) {
		tmp = R * (2.0 * atan2(sqrt(t_3), sqrt((1.0 - fma(0.5, (pow(sin((-0.5 * lambda2)), 2.0) * (cos((phi1 + phi2)) + cos((phi2 - phi1)))), t_3)))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(0.5 * Float64(lambda1 - lambda2))) ^ 2.0
	t_1 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_2 = Float64(R * Float64(2.0 * atan(sqrt(fma(cos(phi1), t_0, Float64(0.25 * (phi1 ^ 2.0)))), sqrt(Float64(1.0 - fma(cos(phi2), t_0, (sin(Float64(-0.5 * phi2)) ^ 2.0)))))))
	t_3 = sin(Float64(0.5 * Float64(phi1 - phi2))) ^ 2.0
	tmp = 0.0
	if (t_1 <= -1e-10)
		tmp = t_2;
	elseif (t_1 <= 1e-42)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_3), sqrt(Float64(1.0 - fma(0.5, Float64((sin(Float64(-0.5 * lambda2)) ^ 2.0) * Float64(cos(Float64(phi1 + phi2)) + cos(Float64(phi2 - phi1)))), t_3))))));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Power[N[Sin[N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[Cos[phi1], $MachinePrecision] * t$95$0 + N[(0.25 * N[Power[phi1, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(N[Cos[phi2], $MachinePrecision] * t$95$0 + N[Power[N[Sin[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Power[N[Sin[N[(0.5 * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[t$95$1, -1e-10], t$95$2, If[LessEqual[t$95$1, 1e-42], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$3], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(0.5 * N[(N[Power[N[Sin[N[(-0.5 * lambda2), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Cos[N[(phi1 + phi2), $MachinePrecision]], $MachinePrecision] + N[Cos[N[(phi2 - phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64))) < -1.00000000000000004e-10 or 1.00000000000000004e-42 < (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))

   1. Initial program 62.0%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-*.f64 46.7

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 46.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   6. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      9. lower-*.f64 46.8

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 46.8%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]

   8. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      2. lower-pow.f64 31.8

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   10. Applied rewrites 31.8%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   11. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}\right) \]
   12. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}\right) \]

       2. lower-pow.f64 22.3

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}\right) \]

   13. Applied rewrites 22.3%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}\right) \]

   14. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{\color{blue}{1 - \left(\cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}}\right) \]
   15. Step-by-step derivation

       1. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}}\right) \]

       2. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       3. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       4. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       5. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       6. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       7. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       8. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       9. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       10. lower-*.f64 32.1

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}\right) \]

   16. Applied rewrites 32.1%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{\color{blue}{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}}\right) \]

   if -1.00000000000000004e-10 < (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64))) < 1.00000000000000004e-42

   1. Initial program 62.0%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in lambda1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\color{blue}{\sqrt{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}\right) + {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. lower-sqrt.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}\right) + {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower--.f64 45.6

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 45.6%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\color{blue}{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right)} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\color{blue}{\cos \phi_1} \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\cos \phi_2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. cos-mult N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\color{blue}{\frac{\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_1 - \phi_2\right)}{2}} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\color{blue}{\frac{1}{\frac{2}{\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_1 - \phi_2\right)}}} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower-unsound-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\color{blue}{\frac{1}{\frac{2}{\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_1 - \phi_2\right)}}} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-unsound-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\frac{1}{\color{blue}{\frac{2}{\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_1 - \phi_2\right)}}} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. +-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\frac{1}{\frac{2}{\color{blue}{\cos \left(\phi_1 - \phi_2\right) + \cos \left(\phi_1 + \phi_2\right)}}} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-+.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\frac{1}{\frac{2}{\color{blue}{\cos \left(\phi_1 - \phi_2\right) + \cos \left(\phi_1 + \phi_2\right)}}} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\frac{1}{\frac{2}{\cos \color{blue}{\left(\phi_1 - \phi_2\right)} + \cos \left(\phi_1 + \phi_2\right)}} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. cos-neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\frac{1}{\frac{2}{\color{blue}{\cos \left(\mathsf{neg}\left(\left(\phi_1 - \phi_2\right)\right)\right)} + \cos \left(\phi_1 + \phi_2\right)}} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\frac{1}{\frac{2}{\color{blue}{\cos \left(\mathsf{neg}\left(\left(\phi_1 - \phi_2\right)\right)\right)} + \cos \left(\phi_1 + \phi_2\right)}} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\frac{1}{\frac{2}{\cos \left(\mathsf{neg}\left(\color{blue}{\left(\phi_1 - \phi_2\right)}\right)\right) + \cos \left(\phi_1 + \phi_2\right)}} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. sub-negate-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\frac{1}{\frac{2}{\cos \color{blue}{\left(\phi_2 - \phi_1\right)} + \cos \left(\phi_1 + \phi_2\right)}} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\frac{1}{\frac{2}{\cos \color{blue}{\left(\phi_2 - \phi_1\right)} + \cos \left(\phi_1 + \phi_2\right)}} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\frac{1}{\frac{2}{\cos \left(\phi_2 - \phi_1\right) + \color{blue}{\cos \left(\phi_1 + \phi_2\right)}}} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. +-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\frac{1}{\frac{2}{\cos \left(\phi_2 - \phi_1\right) + \cos \color{blue}{\left(\phi_2 + \phi_1\right)}}} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-+.f64 46.1

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\frac{1}{\frac{2}{\cos \left(\phi_2 - \phi_1\right) + \cos \color{blue}{\left(\phi_2 + \phi_1\right)}}} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   6. Applied rewrites 46.1%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\color{blue}{\frac{1}{\frac{2}{\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)}}} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Taylor expanded in lambda1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\frac{1}{2} \cdot \left({\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2} \cdot \left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_2 - \phi_1\right)\right)\right) + {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}}\right) \]
   8. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2}, \color{blue}{{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2} \cdot \left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_2 - \phi_1\right)\right)}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2}, {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2} \cdot \color{blue}{\left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_2 - \phi_1\right)\right)}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2}, {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2} \cdot \left(\color{blue}{\cos \left(\phi_1 + \phi_2\right)} + \cos \left(\phi_2 - \phi_1\right)\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2}, {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2} \cdot \left(\cos \color{blue}{\left(\phi_1 + \phi_2\right)} + \cos \left(\phi_2 - \phi_1\right)\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2}, {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2} \cdot \left(\cos \left(\color{blue}{\phi_1} + \phi_2\right) + \cos \left(\phi_2 - \phi_1\right)\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

      6. lower-+.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2}, {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2} \cdot \left(\cos \left(\phi_1 + \phi_2\right) + \color{blue}{\cos \left(\phi_2 - \phi_1\right)}\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

      7. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2}, {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2} \cdot \left(\cos \left(\phi_1 + \phi_2\right) + \cos \color{blue}{\left(\phi_2 - \phi_1\right)}\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

      8. lower-+.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2}, {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2} \cdot \left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\color{blue}{\phi_2} - \phi_1\right)\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

      9. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2}, {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2} \cdot \left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_2 - \phi_1\right)\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

      10. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2}, {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2} \cdot \left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_2 - \phi_1\right)\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

   9. Applied rewrites 45.8%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(0.5, {\sin \left(-0.5 \cdot \lambda_2\right)}^{2} \cdot \left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_2 - \phi_1\right)\right), {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}}\right) \]

   10. Taylor expanded in lambda2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2}, {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2} \cdot \left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_2 - \phi_1\right)\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]
   11. Step-by-step derivation

       1. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2}, {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2} \cdot \left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_2 - \phi_1\right)\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       2. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2}, {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2} \cdot \left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_2 - \phi_1\right)\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2}, {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2} \cdot \left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_2 - \phi_1\right)\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       4. lower--.f64 29.5

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(0.5, {\sin \left(-0.5 \cdot \lambda_2\right)}^{2} \cdot \left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_2 - \phi_1\right)\right), {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

   12. Applied rewrites 29.5%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(0.5, {\sin \left(-0.5 \cdot \lambda_2\right)}^{2} \cdot \left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_2 - \phi_1\right)\right), {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 36: 40.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} t_0 := {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\\ t_1 := \frac{\lambda_1 - \lambda_2}{2}\\ t_2 := {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\\ t_3 := R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, t\_0, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, t\_0, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right)\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-10}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_1 \leq 10^{-42}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_2}}{\sqrt{1 - \mathsf{fma}\left(0.5, {\sin \left(-0.5 \cdot \lambda_2\right)}^{2} \cdot \left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_2 - \phi_1\right)\right), t\_2\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (pow (sin (* 0.5 (- lambda1 lambda2))) 2.0))
        (t_1 (/ (- lambda1 lambda2) 2.0))
        (t_2 (pow (sin (* 0.5 (- phi1 phi2))) 2.0))
        (t_3
         (*
          R
          (*
           2.0
           (atan2
            (sqrt (fma (cos phi1) t_0 (* 0.25 (pow phi1 2.0))))
            (sqrt
             (- 1.0 (fma (cos phi1) t_0 (pow (sin (* 0.5 phi1)) 2.0)))))))))
   (if (<= t_1 -1e-10)
     t_3
     (if (<= t_1 1e-42)
       (*
        R
        (*
         2.0
         (atan2
          (sqrt t_2)
          (sqrt
           (-
            1.0
            (fma
             0.5
             (*
              (pow (sin (* -0.5 lambda2)) 2.0)
              (+ (cos (+ phi1 phi2)) (cos (- phi2 phi1))))
             t_2))))))
       t_3))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = pow(sin((0.5 * (lambda1 - lambda2))), 2.0);
	double t_1 = (lambda1 - lambda2) / 2.0;
	double t_2 = pow(sin((0.5 * (phi1 - phi2))), 2.0);
	double t_3 = R * (2.0 * atan2(sqrt(fma(cos(phi1), t_0, (0.25 * pow(phi1, 2.0)))), sqrt((1.0 - fma(cos(phi1), t_0, pow(sin((0.5 * phi1)), 2.0))))));
	double tmp;
	if (t_1 <= -1e-10) {
		tmp = t_3;
	} else if (t_1 <= 1e-42) {
		tmp = R * (2.0 * atan2(sqrt(t_2), sqrt((1.0 - fma(0.5, (pow(sin((-0.5 * lambda2)), 2.0) * (cos((phi1 + phi2)) + cos((phi2 - phi1)))), t_2)))));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(0.5 * Float64(lambda1 - lambda2))) ^ 2.0
	t_1 = Float64(Float64(lambda1 - lambda2) / 2.0)
	t_2 = sin(Float64(0.5 * Float64(phi1 - phi2))) ^ 2.0
	t_3 = Float64(R * Float64(2.0 * atan(sqrt(fma(cos(phi1), t_0, Float64(0.25 * (phi1 ^ 2.0)))), sqrt(Float64(1.0 - fma(cos(phi1), t_0, (sin(Float64(0.5 * phi1)) ^ 2.0)))))))
	tmp = 0.0
	if (t_1 <= -1e-10)
		tmp = t_3;
	elseif (t_1 <= 1e-42)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_2), sqrt(Float64(1.0 - fma(0.5, Float64((sin(Float64(-0.5 * lambda2)) ^ 2.0) * Float64(cos(Float64(phi1 + phi2)) + cos(Float64(phi2 - phi1)))), t_2))))));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Power[N[Sin[N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[N[(0.5 * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[Cos[phi1], $MachinePrecision] * t$95$0 + N[(0.25 * N[Power[phi1, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(N[Cos[phi1], $MachinePrecision] * t$95$0 + N[Power[N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-10], t$95$3, If[LessEqual[t$95$1, 1e-42], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$2], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(0.5 * N[(N[Power[N[Sin[N[(-0.5 * lambda2), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Cos[N[(phi1 + phi2), $MachinePrecision]], $MachinePrecision] + N[Cos[N[(phi2 - phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)) < -1.00000000000000004e-10 or 1.00000000000000004e-42 < (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64))

   1. Initial program 62.0%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-*.f64 46.7

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 46.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   6. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      9. lower-*.f64 46.8

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 46.8%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]

   8. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      2. lower-pow.f64 31.8

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   10. Applied rewrites 31.8%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   11. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}\right) \]
   12. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}\right) \]

       2. lower-pow.f64 22.3

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}\right) \]

   13. Applied rewrites 22.3%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}\right) \]

   14. Taylor expanded in phi2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{\color{blue}{1 - \left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   15. Step-by-step derivation

       1. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]

       2. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

       3. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

       4. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

       5. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

       6. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

       7. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

       8. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

       9. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

       10. lower-*.f64 31.8

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   16. Applied rewrites 31.8%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{\color{blue}{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]

   if -1.00000000000000004e-10 < (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)) < 1.00000000000000004e-42

   1. Initial program 62.0%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in lambda1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\color{blue}{\sqrt{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}\right) + {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. lower-sqrt.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}\right) + {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower--.f64 45.6

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 45.6%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\color{blue}{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right)} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\color{blue}{\cos \phi_1} \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\cos \phi_2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. cos-mult N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\color{blue}{\frac{\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_1 - \phi_2\right)}{2}} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\color{blue}{\frac{1}{\frac{2}{\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_1 - \phi_2\right)}}} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower-unsound-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\color{blue}{\frac{1}{\frac{2}{\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_1 - \phi_2\right)}}} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-unsound-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\frac{1}{\color{blue}{\frac{2}{\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_1 - \phi_2\right)}}} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. +-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\frac{1}{\frac{2}{\color{blue}{\cos \left(\phi_1 - \phi_2\right) + \cos \left(\phi_1 + \phi_2\right)}}} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-+.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\frac{1}{\frac{2}{\color{blue}{\cos \left(\phi_1 - \phi_2\right) + \cos \left(\phi_1 + \phi_2\right)}}} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\frac{1}{\frac{2}{\cos \color{blue}{\left(\phi_1 - \phi_2\right)} + \cos \left(\phi_1 + \phi_2\right)}} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. cos-neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\frac{1}{\frac{2}{\color{blue}{\cos \left(\mathsf{neg}\left(\left(\phi_1 - \phi_2\right)\right)\right)} + \cos \left(\phi_1 + \phi_2\right)}} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\frac{1}{\frac{2}{\color{blue}{\cos \left(\mathsf{neg}\left(\left(\phi_1 - \phi_2\right)\right)\right)} + \cos \left(\phi_1 + \phi_2\right)}} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\frac{1}{\frac{2}{\cos \left(\mathsf{neg}\left(\color{blue}{\left(\phi_1 - \phi_2\right)}\right)\right) + \cos \left(\phi_1 + \phi_2\right)}} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. sub-negate-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\frac{1}{\frac{2}{\cos \color{blue}{\left(\phi_2 - \phi_1\right)} + \cos \left(\phi_1 + \phi_2\right)}} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\frac{1}{\frac{2}{\cos \color{blue}{\left(\phi_2 - \phi_1\right)} + \cos \left(\phi_1 + \phi_2\right)}} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\frac{1}{\frac{2}{\cos \left(\phi_2 - \phi_1\right) + \color{blue}{\cos \left(\phi_1 + \phi_2\right)}}} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. +-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\frac{1}{\frac{2}{\cos \left(\phi_2 - \phi_1\right) + \cos \color{blue}{\left(\phi_2 + \phi_1\right)}}} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-+.f64 46.1

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\frac{1}{\frac{2}{\cos \left(\phi_2 - \phi_1\right) + \cos \color{blue}{\left(\phi_2 + \phi_1\right)}}} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   6. Applied rewrites 46.1%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\color{blue}{\frac{1}{\frac{2}{\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)}}} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Taylor expanded in lambda1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\frac{1}{2} \cdot \left({\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2} \cdot \left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_2 - \phi_1\right)\right)\right) + {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}}\right) \]
   8. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2}, \color{blue}{{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2} \cdot \left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_2 - \phi_1\right)\right)}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2}, {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2} \cdot \color{blue}{\left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_2 - \phi_1\right)\right)}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2}, {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2} \cdot \left(\color{blue}{\cos \left(\phi_1 + \phi_2\right)} + \cos \left(\phi_2 - \phi_1\right)\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2}, {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2} \cdot \left(\cos \color{blue}{\left(\phi_1 + \phi_2\right)} + \cos \left(\phi_2 - \phi_1\right)\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2}, {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2} \cdot \left(\cos \left(\color{blue}{\phi_1} + \phi_2\right) + \cos \left(\phi_2 - \phi_1\right)\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

      6. lower-+.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2}, {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2} \cdot \left(\cos \left(\phi_1 + \phi_2\right) + \color{blue}{\cos \left(\phi_2 - \phi_1\right)}\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

      7. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2}, {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2} \cdot \left(\cos \left(\phi_1 + \phi_2\right) + \cos \color{blue}{\left(\phi_2 - \phi_1\right)}\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

      8. lower-+.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2}, {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2} \cdot \left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\color{blue}{\phi_2} - \phi_1\right)\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

      9. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2}, {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2} \cdot \left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_2 - \phi_1\right)\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

      10. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2}, {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2} \cdot \left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_2 - \phi_1\right)\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

   9. Applied rewrites 45.8%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(0.5, {\sin \left(-0.5 \cdot \lambda_2\right)}^{2} \cdot \left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_2 - \phi_1\right)\right), {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}}\right) \]

   10. Taylor expanded in lambda2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2}, {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2} \cdot \left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_2 - \phi_1\right)\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]
   11. Step-by-step derivation

       1. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2}, {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2} \cdot \left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_2 - \phi_1\right)\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       2. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2}, {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2} \cdot \left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_2 - \phi_1\right)\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2}, {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2} \cdot \left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_2 - \phi_1\right)\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       4. lower--.f64 29.5

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(0.5, {\sin \left(-0.5 \cdot \lambda_2\right)}^{2} \cdot \left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_2 - \phi_1\right)\right), {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

   12. Applied rewrites 29.5%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(0.5, {\sin \left(-0.5 \cdot \lambda_2\right)}^{2} \cdot \left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_2 - \phi_1\right)\right), {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 37: 39.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_0 := {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\\ t_1 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_2 := \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot 0.25\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}} \cdot 2\right) \cdot R\\ \mathbf{if}\;t\_1 \leq -0.15:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-10}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_0}}{\sqrt{1 - \mathsf{fma}\left(0.5, {\sin \left(-0.5 \cdot \lambda_2\right)}^{2} \cdot \left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_2 - \phi_1\right)\right), t\_0\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (pow (sin (* 0.5 (- phi1 phi2))) 2.0))
        (t_1 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_2
         (*
          (*
           (atan2
            (sqrt
             (fma
              (- 0.5 (* (cos (* 1.0 (- lambda1 lambda2))) 0.5))
              (cos phi1)
              (* (* phi1 phi1) 0.25)))
            (sqrt
             (-
              1.0
              (fma
               (cos phi2)
               (pow (sin (* 0.5 (- lambda1 lambda2))) 2.0)
               (pow (sin (* -0.5 phi2)) 2.0)))))
           2.0)
          R)))
   (if (<= t_1 -0.15)
     t_2
     (if (<= t_1 2e-10)
       (*
        R
        (*
         2.0
         (atan2
          (sqrt t_0)
          (sqrt
           (-
            1.0
            (fma
             0.5
             (*
              (pow (sin (* -0.5 lambda2)) 2.0)
              (+ (cos (+ phi1 phi2)) (cos (- phi2 phi1))))
             t_0))))))
       t_2))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = pow(sin((0.5 * (phi1 - phi2))), 2.0);
	double t_1 = sin(((lambda1 - lambda2) / 2.0));
	double t_2 = (atan2(sqrt(fma((0.5 - (cos((1.0 * (lambda1 - lambda2))) * 0.5)), cos(phi1), ((phi1 * phi1) * 0.25))), sqrt((1.0 - fma(cos(phi2), pow(sin((0.5 * (lambda1 - lambda2))), 2.0), pow(sin((-0.5 * phi2)), 2.0))))) * 2.0) * R;
	double tmp;
	if (t_1 <= -0.15) {
		tmp = t_2;
	} else if (t_1 <= 2e-10) {
		tmp = R * (2.0 * atan2(sqrt(t_0), sqrt((1.0 - fma(0.5, (pow(sin((-0.5 * lambda2)), 2.0) * (cos((phi1 + phi2)) + cos((phi2 - phi1)))), t_0)))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(0.5 * Float64(phi1 - phi2))) ^ 2.0
	t_1 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_2 = Float64(Float64(atan(sqrt(fma(Float64(0.5 - Float64(cos(Float64(1.0 * Float64(lambda1 - lambda2))) * 0.5)), cos(phi1), Float64(Float64(phi1 * phi1) * 0.25))), sqrt(Float64(1.0 - fma(cos(phi2), (sin(Float64(0.5 * Float64(lambda1 - lambda2))) ^ 2.0), (sin(Float64(-0.5 * phi2)) ^ 2.0))))) * 2.0) * R)
	tmp = 0.0
	if (t_1 <= -0.15)
		tmp = t_2;
	elseif (t_1 <= 2e-10)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_0), sqrt(Float64(1.0 - fma(0.5, Float64((sin(Float64(-0.5 * lambda2)) ^ 2.0) * Float64(cos(Float64(phi1 + phi2)) + cos(Float64(phi2 - phi1)))), t_0))))));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Power[N[Sin[N[(0.5 * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[ArcTan[N[Sqrt[N[(N[(0.5 - N[(N[Cos[N[(1.0 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision] + N[(N[(phi1 * phi1), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(N[Cos[phi2], $MachinePrecision] * N[Power[N[Sin[N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * R), $MachinePrecision]}, If[LessEqual[t$95$1, -0.15], t$95$2, If[LessEqual[t$95$1, 2e-10], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$0], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(0.5 * N[(N[Power[N[Sin[N[(-0.5 * lambda2), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Cos[N[(phi1 + phi2), $MachinePrecision]], $MachinePrecision] + N[Cos[N[(phi2 - phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64))) < -0.149999999999999994 or 2.00000000000000007e-10 < (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))

   1. Initial program 62.0%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-*.f64 46.7

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 46.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   6. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      9. lower-*.f64 46.8

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 46.8%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]

   8. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      2. lower-pow.f64 31.8

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   10. Applied rewrites 31.8%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   11. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}\right) \]
   12. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}\right) \]

       2. lower-pow.f64 22.3

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}\right) \]

   13. Applied rewrites 22.3%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}\right) \]

   15. Applied rewrites 19.9%

       \[\leadsto \color{blue}{\left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot 0.25\right)}}{\sqrt{1 - \mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot 0.25\right)}} \cdot 2\right) \cdot R} \]

   16. Taylor expanded in phi1 around 0

       \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}}{\sqrt{\color{blue}{1 - \left(\cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}} \cdot 2\right) \cdot R \]
   17. Step-by-step derivation

       1. lower--.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}} \cdot 2\right) \cdot R \]

       2. lower-fma.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}} \cdot 2\right) \cdot R \]

       3. lower-cos.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}} \cdot 2\right) \cdot R \]

       4. lower-pow.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}} \cdot 2\right) \cdot R \]

       5. lower-sin.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}} \cdot 2\right) \cdot R \]

       6. lower-*.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}} \cdot 2\right) \cdot R \]

       7. lower--.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}} \cdot 2\right) \cdot R \]

       8. lower-pow.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}} \cdot 2\right) \cdot R \]

       9. lower-sin.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}} \cdot 2\right) \cdot R \]

       10. lower-*.f64 29.6

           \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot 0.25\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}} \cdot 2\right) \cdot R \]

   18. Applied rewrites 29.6%

       \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot 0.25\right)}}{\sqrt{\color{blue}{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}} \cdot 2\right) \cdot R \]

   if -0.149999999999999994 < (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64))) < 2.00000000000000007e-10

   1. Initial program 62.0%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in lambda1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\color{blue}{\sqrt{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}\right) + {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. lower-sqrt.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}\right) + {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower--.f64 45.6

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 45.6%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\color{blue}{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right)} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\color{blue}{\cos \phi_1} \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\cos \phi_2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. cos-mult N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\color{blue}{\frac{\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_1 - \phi_2\right)}{2}} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\color{blue}{\frac{1}{\frac{2}{\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_1 - \phi_2\right)}}} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower-unsound-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\color{blue}{\frac{1}{\frac{2}{\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_1 - \phi_2\right)}}} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-unsound-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\frac{1}{\color{blue}{\frac{2}{\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_1 - \phi_2\right)}}} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. +-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\frac{1}{\frac{2}{\color{blue}{\cos \left(\phi_1 - \phi_2\right) + \cos \left(\phi_1 + \phi_2\right)}}} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-+.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\frac{1}{\frac{2}{\color{blue}{\cos \left(\phi_1 - \phi_2\right) + \cos \left(\phi_1 + \phi_2\right)}}} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\frac{1}{\frac{2}{\cos \color{blue}{\left(\phi_1 - \phi_2\right)} + \cos \left(\phi_1 + \phi_2\right)}} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. cos-neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\frac{1}{\frac{2}{\color{blue}{\cos \left(\mathsf{neg}\left(\left(\phi_1 - \phi_2\right)\right)\right)} + \cos \left(\phi_1 + \phi_2\right)}} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\frac{1}{\frac{2}{\color{blue}{\cos \left(\mathsf{neg}\left(\left(\phi_1 - \phi_2\right)\right)\right)} + \cos \left(\phi_1 + \phi_2\right)}} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\frac{1}{\frac{2}{\cos \left(\mathsf{neg}\left(\color{blue}{\left(\phi_1 - \phi_2\right)}\right)\right) + \cos \left(\phi_1 + \phi_2\right)}} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. sub-negate-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\frac{1}{\frac{2}{\cos \color{blue}{\left(\phi_2 - \phi_1\right)} + \cos \left(\phi_1 + \phi_2\right)}} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\frac{1}{\frac{2}{\cos \color{blue}{\left(\phi_2 - \phi_1\right)} + \cos \left(\phi_1 + \phi_2\right)}} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\frac{1}{\frac{2}{\cos \left(\phi_2 - \phi_1\right) + \color{blue}{\cos \left(\phi_1 + \phi_2\right)}}} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. +-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\frac{1}{\frac{2}{\cos \left(\phi_2 - \phi_1\right) + \cos \color{blue}{\left(\phi_2 + \phi_1\right)}}} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-+.f64 46.1

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\frac{1}{\frac{2}{\cos \left(\phi_2 - \phi_1\right) + \cos \color{blue}{\left(\phi_2 + \phi_1\right)}}} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   6. Applied rewrites 46.1%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\color{blue}{\frac{1}{\frac{2}{\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)}}} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Taylor expanded in lambda1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\frac{1}{2} \cdot \left({\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2} \cdot \left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_2 - \phi_1\right)\right)\right) + {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}}\right) \]
   8. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2}, \color{blue}{{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2} \cdot \left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_2 - \phi_1\right)\right)}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2}, {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2} \cdot \color{blue}{\left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_2 - \phi_1\right)\right)}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2}, {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2} \cdot \left(\color{blue}{\cos \left(\phi_1 + \phi_2\right)} + \cos \left(\phi_2 - \phi_1\right)\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2}, {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2} \cdot \left(\cos \color{blue}{\left(\phi_1 + \phi_2\right)} + \cos \left(\phi_2 - \phi_1\right)\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2}, {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2} \cdot \left(\cos \left(\color{blue}{\phi_1} + \phi_2\right) + \cos \left(\phi_2 - \phi_1\right)\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

      6. lower-+.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2}, {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2} \cdot \left(\cos \left(\phi_1 + \phi_2\right) + \color{blue}{\cos \left(\phi_2 - \phi_1\right)}\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

      7. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2}, {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2} \cdot \left(\cos \left(\phi_1 + \phi_2\right) + \cos \color{blue}{\left(\phi_2 - \phi_1\right)}\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

      8. lower-+.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2}, {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2} \cdot \left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\color{blue}{\phi_2} - \phi_1\right)\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

      9. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2}, {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2} \cdot \left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_2 - \phi_1\right)\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

      10. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2}, {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2} \cdot \left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_2 - \phi_1\right)\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

   9. Applied rewrites 45.8%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(0.5, {\sin \left(-0.5 \cdot \lambda_2\right)}^{2} \cdot \left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_2 - \phi_1\right)\right), {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}}\right) \]

   10. Taylor expanded in lambda2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2}, {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2} \cdot \left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_2 - \phi_1\right)\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]
   11. Step-by-step derivation

       1. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2}, {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2} \cdot \left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_2 - \phi_1\right)\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       2. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2}, {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2} \cdot \left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_2 - \phi_1\right)\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2}, {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2} \cdot \left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_2 - \phi_1\right)\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       4. lower--.f64 29.5

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(0.5, {\sin \left(-0.5 \cdot \lambda_2\right)}^{2} \cdot \left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_2 - \phi_1\right)\right), {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

   12. Applied rewrites 29.5%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(0.5, {\sin \left(-0.5 \cdot \lambda_2\right)}^{2} \cdot \left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_2 - \phi_1\right)\right), {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 38: 33.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot 0.25\right)\\ t_1 := \mathsf{fma}\left(\cos \phi_2, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)\\ t_2 := \left(\tan^{-1}_* \frac{\sqrt{t\_0}}{\sqrt{1 - t\_1}} \cdot 2\right) \cdot R\\ \mathbf{if}\;\phi_1 \leq -2 \cdot 10^{-57}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\phi_1 \leq 7 \cdot 10^{-40}:\\ \;\;\;\;\left(\tan^{-1}_* \frac{\sqrt{t\_1}}{\sqrt{1 - t\_0}} \cdot 2\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (fma
          (- 0.5 (* (cos (* 1.0 (- lambda1 lambda2))) 0.5))
          (cos phi1)
          (* (* phi1 phi1) 0.25)))
        (t_1
         (fma
          (cos phi2)
          (pow (sin (* 0.5 (- lambda1 lambda2))) 2.0)
          (pow (sin (* -0.5 phi2)) 2.0)))
        (t_2 (* (* (atan2 (sqrt t_0) (sqrt (- 1.0 t_1))) 2.0) R)))
   (if (<= phi1 -2e-57)
     t_2
     (if (<= phi1 7e-40)
       (* (* (atan2 (sqrt t_1) (sqrt (- 1.0 t_0))) 2.0) R)
       t_2))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = fma((0.5 - (cos((1.0 * (lambda1 - lambda2))) * 0.5)), cos(phi1), ((phi1 * phi1) * 0.25));
	double t_1 = fma(cos(phi2), pow(sin((0.5 * (lambda1 - lambda2))), 2.0), pow(sin((-0.5 * phi2)), 2.0));
	double t_2 = (atan2(sqrt(t_0), sqrt((1.0 - t_1))) * 2.0) * R;
	double tmp;
	if (phi1 <= -2e-57) {
		tmp = t_2;
	} else if (phi1 <= 7e-40) {
		tmp = (atan2(sqrt(t_1), sqrt((1.0 - t_0))) * 2.0) * R;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = fma(Float64(0.5 - Float64(cos(Float64(1.0 * Float64(lambda1 - lambda2))) * 0.5)), cos(phi1), Float64(Float64(phi1 * phi1) * 0.25))
	t_1 = fma(cos(phi2), (sin(Float64(0.5 * Float64(lambda1 - lambda2))) ^ 2.0), (sin(Float64(-0.5 * phi2)) ^ 2.0))
	t_2 = Float64(Float64(atan(sqrt(t_0), sqrt(Float64(1.0 - t_1))) * 2.0) * R)
	tmp = 0.0
	if (phi1 <= -2e-57)
		tmp = t_2;
	elseif (phi1 <= 7e-40)
		tmp = Float64(Float64(atan(sqrt(t_1), sqrt(Float64(1.0 - t_0))) * 2.0) * R);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[(0.5 - N[(N[Cos[N[(1.0 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision] + N[(N[(phi1 * phi1), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[phi2], $MachinePrecision] * N[Power[N[Sin[N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[ArcTan[N[Sqrt[t$95$0], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * R), $MachinePrecision]}, If[LessEqual[phi1, -2e-57], t$95$2, If[LessEqual[phi1, 7e-40], N[(N[(N[ArcTan[N[Sqrt[t$95$1], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * R), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -1.99999999999999991e-57 or 7.0000000000000003e-40 < phi1

   1. Initial program 62.0%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-*.f64 46.7

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 46.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   6. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      9. lower-*.f64 46.8

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 46.8%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]

   8. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      2. lower-pow.f64 31.8

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   10. Applied rewrites 31.8%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   11. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}\right) \]
   12. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}\right) \]

       2. lower-pow.f64 22.3

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}\right) \]

   13. Applied rewrites 22.3%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}\right) \]

   15. Applied rewrites 19.9%

       \[\leadsto \color{blue}{\left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot 0.25\right)}}{\sqrt{1 - \mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot 0.25\right)}} \cdot 2\right) \cdot R} \]

   16. Taylor expanded in phi1 around 0

       \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}}{\sqrt{\color{blue}{1 - \left(\cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}} \cdot 2\right) \cdot R \]
   17. Step-by-step derivation

       1. lower--.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}} \cdot 2\right) \cdot R \]

       2. lower-fma.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}} \cdot 2\right) \cdot R \]

       3. lower-cos.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}} \cdot 2\right) \cdot R \]

       4. lower-pow.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}} \cdot 2\right) \cdot R \]

       5. lower-sin.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}} \cdot 2\right) \cdot R \]

       6. lower-*.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}} \cdot 2\right) \cdot R \]

       7. lower--.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}} \cdot 2\right) \cdot R \]

       8. lower-pow.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}} \cdot 2\right) \cdot R \]

       9. lower-sin.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}} \cdot 2\right) \cdot R \]

       10. lower-*.f64 29.6

           \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot 0.25\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}} \cdot 2\right) \cdot R \]

   18. Applied rewrites 29.6%

       \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot 0.25\right)}}{\sqrt{\color{blue}{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}} \cdot 2\right) \cdot R \]

   if -1.99999999999999991e-57 < phi1 < 7.0000000000000003e-40

   1. Initial program 62.0%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-*.f64 46.7

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 46.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   6. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      9. lower-*.f64 46.8

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 46.8%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]

   8. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      2. lower-pow.f64 31.8

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   10. Applied rewrites 31.8%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   11. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}\right) \]
   12. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}\right) \]

       2. lower-pow.f64 22.3

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}\right) \]

   13. Applied rewrites 22.3%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}\right) \]

   15. Applied rewrites 19.9%

       \[\leadsto \color{blue}{\left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot 0.25\right)}}{\sqrt{1 - \mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot 0.25\right)}} \cdot 2\right) \cdot R} \]

   16. Taylor expanded in phi1 around 0

       \[\leadsto \left(\tan^{-1}_* \frac{\color{blue}{\sqrt{\cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}}}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}} \cdot 2\right) \cdot R \]
   17. Step-by-step derivation

       1. lower-sqrt.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}} \cdot 2\right) \cdot R \]

       2. lower-fma.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}} \cdot 2\right) \cdot R \]

       3. lower-cos.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}} \cdot 2\right) \cdot R \]

       4. lower-pow.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}} \cdot 2\right) \cdot R \]

       5. lower-sin.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}} \cdot 2\right) \cdot R \]

       6. lower-*.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}} \cdot 2\right) \cdot R \]

       7. lower--.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}} \cdot 2\right) \cdot R \]

       8. lower-pow.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}} \cdot 2\right) \cdot R \]

       9. lower-sin.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}} \cdot 2\right) \cdot R \]

       10. lower-*.f64 23.6

           \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot 0.25\right)}} \cdot 2\right) \cdot R \]

   18. Applied rewrites 23.6%

       \[\leadsto \left(\tan^{-1}_* \frac{\color{blue}{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}}{\sqrt{1 - \mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot 0.25\right)}} \cdot 2\right) \cdot R \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 39: 31.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_0 := {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\\ t_1 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_2 := \left(\phi_1 \cdot \phi_1\right) \cdot 0.25\\ t_3 := \mathsf{fma}\left(\cos \phi_1, t\_0, t\_2\right)\\ t_4 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_1\right) \cdot t\_1\\ \mathbf{if}\;2 \cdot \tan^{-1}_* \frac{\sqrt{t\_4}}{\sqrt{1 - t\_4}} \leq 0.004:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_3}}{\sqrt{1 - t\_3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, \cos \phi_1, t\_2\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, t\_0, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}} \cdot 2\right) \cdot R\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (pow (sin (* 0.5 (- lambda1 lambda2))) 2.0))
        (t_1 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_2 (* (* phi1 phi1) 0.25))
        (t_3 (fma (cos phi1) t_0 t_2))
        (t_4
         (+
          (pow (sin (/ (- phi1 phi2) 2.0)) 2.0)
          (* (* (* (cos phi1) (cos phi2)) t_1) t_1))))
   (if (<= (* 2.0 (atan2 (sqrt t_4) (sqrt (- 1.0 t_4)))) 0.004)
     (* R (* 2.0 (atan2 (sqrt t_3) (sqrt (- 1.0 t_3)))))
     (*
      (*
       (atan2
        (sqrt
         (fma
          (- 0.5 (* (cos (* 1.0 (- lambda1 lambda2))) 0.5))
          (cos phi1)
          t_2))
        (sqrt (- 1.0 (fma (cos phi2) t_0 (pow (sin (* -0.5 phi2)) 2.0)))))
       2.0)
      R))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = pow(sin((0.5 * (lambda1 - lambda2))), 2.0);
	double t_1 = sin(((lambda1 - lambda2) / 2.0));
	double t_2 = (phi1 * phi1) * 0.25;
	double t_3 = fma(cos(phi1), t_0, t_2);
	double t_4 = pow(sin(((phi1 - phi2) / 2.0)), 2.0) + (((cos(phi1) * cos(phi2)) * t_1) * t_1);
	double tmp;
	if ((2.0 * atan2(sqrt(t_4), sqrt((1.0 - t_4)))) <= 0.004) {
		tmp = R * (2.0 * atan2(sqrt(t_3), sqrt((1.0 - t_3))));
	} else {
		tmp = (atan2(sqrt(fma((0.5 - (cos((1.0 * (lambda1 - lambda2))) * 0.5)), cos(phi1), t_2)), sqrt((1.0 - fma(cos(phi2), t_0, pow(sin((-0.5 * phi2)), 2.0))))) * 2.0) * R;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(0.5 * Float64(lambda1 - lambda2))) ^ 2.0
	t_1 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_2 = Float64(Float64(phi1 * phi1) * 0.25)
	t_3 = fma(cos(phi1), t_0, t_2)
	t_4 = Float64((sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0) + Float64(Float64(Float64(cos(phi1) * cos(phi2)) * t_1) * t_1))
	tmp = 0.0
	if (Float64(2.0 * atan(sqrt(t_4), sqrt(Float64(1.0 - t_4)))) <= 0.004)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_3), sqrt(Float64(1.0 - t_3)))));
	else
		tmp = Float64(Float64(atan(sqrt(fma(Float64(0.5 - Float64(cos(Float64(1.0 * Float64(lambda1 - lambda2))) * 0.5)), cos(phi1), t_2)), sqrt(Float64(1.0 - fma(cos(phi2), t_0, (sin(Float64(-0.5 * phi2)) ^ 2.0))))) * 2.0) * R);
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Power[N[Sin[N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(phi1 * phi1), $MachinePrecision] * 0.25), $MachinePrecision]}, Block[{t$95$3 = N[(N[Cos[phi1], $MachinePrecision] * t$95$0 + t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[(N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(2.0 * N[ArcTan[N[Sqrt[t$95$4], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$4), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.004], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$3], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$3), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[ArcTan[N[Sqrt[N[(N[(0.5 - N[(N[Cos[N[(1.0 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision] + t$95$2), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(N[Cos[phi2], $MachinePrecision] * t$95$0 + N[Power[N[Sin[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * R), $MachinePrecision]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 2 binary64) (atan2.f64 (sqrt.f64 (+.f64 (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) #s(literal 2 binary64))) #s(literal 2 binary64)) (*.f64 (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))))) (sqrt.f64 (-.f64 #s(literal 1 binary64) (+.f64 (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) #s(literal 2 binary64))) #s(literal 2 binary64)) (*.f64 (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64))))))))) < 0.0040000000000000001

   1. Initial program 62.0%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-*.f64 46.7

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 46.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   6. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      9. lower-*.f64 46.8

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 46.8%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]

   8. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      2. lower-pow.f64 31.8

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   10. Applied rewrites 31.8%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   11. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}\right) \]
   12. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}\right) \]

       2. lower-pow.f64 22.3

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}\right) \]

   13. Applied rewrites 22.3%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}\right) \]
   14. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}\right) \]

       2. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\phi_1}^{2} \cdot \frac{1}{4}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}\right) \]

       3. lift-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\phi_1}^{2} \cdot \frac{1}{4}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}\right) \]

       4. unpow2 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}\right) \]

       5. lower-*.f32 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}\right) \]

       6. lower-unsound-*.f32 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}\right) \]

       8. lower-unsound-*.f64 22.3

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \left(\phi_1 \cdot \phi_1\right) \cdot 0.25\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}\right) \]

   15. Applied rewrites 22.3%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \left(\phi_1 \cdot \phi_1\right) \cdot 0.25\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}\right) \]
   16. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}\right) \]

       2. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\phi_1}^{2} \cdot \frac{1}{4}\right)}}\right) \]

       3. lift-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\phi_1}^{2} \cdot \frac{1}{4}\right)}}\right) \]

       4. unpow2 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}}\right) \]

       5. lower-*.f32 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}}\right) \]

       6. lower-unsound-*.f32 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}}\right) \]

       8. lower-unsound-*.f64 22.3

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \left(\phi_1 \cdot \phi_1\right) \cdot 0.25\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \left(\phi_1 \cdot \phi_1\right) \cdot 0.25\right)}}\right) \]

   17. Applied rewrites 22.3%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \left(\phi_1 \cdot \phi_1\right) \cdot 0.25\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \left(\phi_1 \cdot \phi_1\right) \cdot 0.25\right)}}\right) \]

   if 0.0040000000000000001 < (*.f64 #s(literal 2 binary64) (atan2.f64 (sqrt.f64 (+.f64 (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) #s(literal 2 binary64))) #s(literal 2 binary64)) (*.f64 (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))))) (sqrt.f64 (-.f64 #s(literal 1 binary64) (+.f64 (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) #s(literal 2 binary64))) #s(literal 2 binary64)) (*.f64 (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))))))))

   1. Initial program 62.0%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-*.f64 46.7

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 46.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   6. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      9. lower-*.f64 46.8

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 46.8%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]

   8. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      2. lower-pow.f64 31.8

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   10. Applied rewrites 31.8%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   11. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}\right) \]
   12. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}\right) \]

       2. lower-pow.f64 22.3

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}\right) \]

   13. Applied rewrites 22.3%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}\right) \]

   15. Applied rewrites 19.9%

       \[\leadsto \color{blue}{\left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot 0.25\right)}}{\sqrt{1 - \mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot 0.25\right)}} \cdot 2\right) \cdot R} \]

   16. Taylor expanded in phi1 around 0

       \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}}{\sqrt{\color{blue}{1 - \left(\cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}} \cdot 2\right) \cdot R \]
   17. Step-by-step derivation

       1. lower--.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}} \cdot 2\right) \cdot R \]

       2. lower-fma.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}} \cdot 2\right) \cdot R \]

       3. lower-cos.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}} \cdot 2\right) \cdot R \]

       4. lower-pow.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}} \cdot 2\right) \cdot R \]

       5. lower-sin.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}} \cdot 2\right) \cdot R \]

       6. lower-*.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}} \cdot 2\right) \cdot R \]

       7. lower--.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}} \cdot 2\right) \cdot R \]

       8. lower-pow.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}} \cdot 2\right) \cdot R \]

       9. lower-sin.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}} \cdot 2\right) \cdot R \]

       10. lower-*.f64 29.6

           \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot 0.25\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}} \cdot 2\right) \cdot R \]

   18. Applied rewrites 29.6%

       \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot 0.25\right)}}{\sqrt{\color{blue}{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}} \cdot 2\right) \cdot R \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 40: 31.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_0 := {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\\ t_1 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_2 := \left(\phi_1 \cdot \phi_1\right) \cdot 0.25\\ t_3 := \mathsf{fma}\left(\cos \phi_1, t\_0, t\_2\right)\\ t_4 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_1\right) \cdot t\_1\\ \mathbf{if}\;2 \cdot \tan^{-1}_* \frac{\sqrt{t\_4}}{\sqrt{1 - t\_4}} \leq 0.004:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_3}}{\sqrt{1 - t\_3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, \cos \phi_1, t\_2\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, t\_0, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}} \cdot 2\right) \cdot R\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (pow (sin (* 0.5 (- lambda1 lambda2))) 2.0))
        (t_1 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_2 (* (* phi1 phi1) 0.25))
        (t_3 (fma (cos phi1) t_0 t_2))
        (t_4
         (+
          (pow (sin (/ (- phi1 phi2) 2.0)) 2.0)
          (* (* (* (cos phi1) (cos phi2)) t_1) t_1))))
   (if (<= (* 2.0 (atan2 (sqrt t_4) (sqrt (- 1.0 t_4)))) 0.004)
     (* R (* 2.0 (atan2 (sqrt t_3) (sqrt (- 1.0 t_3)))))
     (*
      (*
       (atan2
        (sqrt
         (fma
          (- 0.5 (* (cos (* 1.0 (- lambda1 lambda2))) 0.5))
          (cos phi1)
          t_2))
        (sqrt (- 1.0 (fma (cos phi1) t_0 (pow (sin (* 0.5 phi1)) 2.0)))))
       2.0)
      R))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = pow(sin((0.5 * (lambda1 - lambda2))), 2.0);
	double t_1 = sin(((lambda1 - lambda2) / 2.0));
	double t_2 = (phi1 * phi1) * 0.25;
	double t_3 = fma(cos(phi1), t_0, t_2);
	double t_4 = pow(sin(((phi1 - phi2) / 2.0)), 2.0) + (((cos(phi1) * cos(phi2)) * t_1) * t_1);
	double tmp;
	if ((2.0 * atan2(sqrt(t_4), sqrt((1.0 - t_4)))) <= 0.004) {
		tmp = R * (2.0 * atan2(sqrt(t_3), sqrt((1.0 - t_3))));
	} else {
		tmp = (atan2(sqrt(fma((0.5 - (cos((1.0 * (lambda1 - lambda2))) * 0.5)), cos(phi1), t_2)), sqrt((1.0 - fma(cos(phi1), t_0, pow(sin((0.5 * phi1)), 2.0))))) * 2.0) * R;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(0.5 * Float64(lambda1 - lambda2))) ^ 2.0
	t_1 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_2 = Float64(Float64(phi1 * phi1) * 0.25)
	t_3 = fma(cos(phi1), t_0, t_2)
	t_4 = Float64((sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0) + Float64(Float64(Float64(cos(phi1) * cos(phi2)) * t_1) * t_1))
	tmp = 0.0
	if (Float64(2.0 * atan(sqrt(t_4), sqrt(Float64(1.0 - t_4)))) <= 0.004)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_3), sqrt(Float64(1.0 - t_3)))));
	else
		tmp = Float64(Float64(atan(sqrt(fma(Float64(0.5 - Float64(cos(Float64(1.0 * Float64(lambda1 - lambda2))) * 0.5)), cos(phi1), t_2)), sqrt(Float64(1.0 - fma(cos(phi1), t_0, (sin(Float64(0.5 * phi1)) ^ 2.0))))) * 2.0) * R);
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Power[N[Sin[N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(phi1 * phi1), $MachinePrecision] * 0.25), $MachinePrecision]}, Block[{t$95$3 = N[(N[Cos[phi1], $MachinePrecision] * t$95$0 + t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[(N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(2.0 * N[ArcTan[N[Sqrt[t$95$4], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$4), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.004], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$3], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$3), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[ArcTan[N[Sqrt[N[(N[(0.5 - N[(N[Cos[N[(1.0 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision] + t$95$2), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(N[Cos[phi1], $MachinePrecision] * t$95$0 + N[Power[N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * R), $MachinePrecision]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 2 binary64) (atan2.f64 (sqrt.f64 (+.f64 (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) #s(literal 2 binary64))) #s(literal 2 binary64)) (*.f64 (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))))) (sqrt.f64 (-.f64 #s(literal 1 binary64) (+.f64 (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) #s(literal 2 binary64))) #s(literal 2 binary64)) (*.f64 (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64))))))))) < 0.0040000000000000001

   1. Initial program 62.0%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-*.f64 46.7

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 46.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   6. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      9. lower-*.f64 46.8

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 46.8%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]

   8. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      2. lower-pow.f64 31.8

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   10. Applied rewrites 31.8%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   11. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}\right) \]
   12. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}\right) \]

       2. lower-pow.f64 22.3

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}\right) \]

   13. Applied rewrites 22.3%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}\right) \]
   14. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}\right) \]

       2. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\phi_1}^{2} \cdot \frac{1}{4}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}\right) \]

       3. lift-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\phi_1}^{2} \cdot \frac{1}{4}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}\right) \]

       4. unpow2 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}\right) \]

       5. lower-*.f32 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}\right) \]

       6. lower-unsound-*.f32 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}\right) \]

       8. lower-unsound-*.f64 22.3

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \left(\phi_1 \cdot \phi_1\right) \cdot 0.25\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}\right) \]

   15. Applied rewrites 22.3%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \left(\phi_1 \cdot \phi_1\right) \cdot 0.25\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}\right) \]
   16. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}\right) \]

       2. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\phi_1}^{2} \cdot \frac{1}{4}\right)}}\right) \]

       3. lift-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\phi_1}^{2} \cdot \frac{1}{4}\right)}}\right) \]

       4. unpow2 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}}\right) \]

       5. lower-*.f32 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}}\right) \]

       6. lower-unsound-*.f32 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}}\right) \]

       8. lower-unsound-*.f64 22.3

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \left(\phi_1 \cdot \phi_1\right) \cdot 0.25\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \left(\phi_1 \cdot \phi_1\right) \cdot 0.25\right)}}\right) \]

   17. Applied rewrites 22.3%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \left(\phi_1 \cdot \phi_1\right) \cdot 0.25\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \left(\phi_1 \cdot \phi_1\right) \cdot 0.25\right)}}\right) \]

   if 0.0040000000000000001 < (*.f64 #s(literal 2 binary64) (atan2.f64 (sqrt.f64 (+.f64 (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) #s(literal 2 binary64))) #s(literal 2 binary64)) (*.f64 (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))))) (sqrt.f64 (-.f64 #s(literal 1 binary64) (+.f64 (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) #s(literal 2 binary64))) #s(literal 2 binary64)) (*.f64 (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))))))))

   1. Initial program 62.0%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-*.f64 46.7

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 46.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   6. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      9. lower-*.f64 46.8

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 46.8%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]

   8. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      2. lower-pow.f64 31.8

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   10. Applied rewrites 31.8%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   11. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}\right) \]
   12. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}\right) \]

       2. lower-pow.f64 22.3

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}\right) \]

   13. Applied rewrites 22.3%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}\right) \]

   15. Applied rewrites 19.9%

       \[\leadsto \color{blue}{\left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot 0.25\right)}}{\sqrt{1 - \mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot 0.25\right)}} \cdot 2\right) \cdot R} \]

   16. Taylor expanded in phi2 around 0

       \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}}{\sqrt{\color{blue}{1 - \left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}} \cdot 2\right) \cdot R \]
   17. Step-by-step derivation

       1. lower--.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}} \cdot 2\right) \cdot R \]

       2. lower-fma.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}} \cdot 2\right) \cdot R \]

       3. lower-cos.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}} \cdot 2\right) \cdot R \]

       4. lower-pow.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}} \cdot 2\right) \cdot R \]

       5. lower-sin.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}} \cdot 2\right) \cdot R \]

       6. lower-*.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}} \cdot 2\right) \cdot R \]

       7. lower--.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}} \cdot 2\right) \cdot R \]

       8. lower-pow.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}} \cdot 2\right) \cdot R \]

       9. lower-sin.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}} \cdot 2\right) \cdot R \]

       10. lower-*.f64 29.4

           \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot 0.25\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}} \cdot 2\right) \cdot R \]

   18. Applied rewrites 29.4%

       \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot 0.25\right)}}{\sqrt{\color{blue}{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}} \cdot 2\right) \cdot R \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 41: 27.5% accurate, 1.8× speedup

Math

\[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{{\left(1 - \mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot 0.25\right)\right)}^{0.5}}\right) \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  R
  (*
   2.0
   (atan2
    (sqrt
     (fma
      (cos phi1)
      (pow (sin (* 0.5 (- lambda1 lambda2))) 2.0)
      (* 0.25 (pow phi1 2.0))))
    (pow
     (-
      1.0
      (fma
       (- 0.5 (* (cos (* 1.0 (- lambda1 lambda2))) 0.5))
       (cos phi1)
       (* (* phi1 phi1) 0.25)))
     0.5)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return R * (2.0 * atan2(sqrt(fma(cos(phi1), pow(sin((0.5 * (lambda1 - lambda2))), 2.0), (0.25 * pow(phi1, 2.0)))), pow((1.0 - fma((0.5 - (cos((1.0 * (lambda1 - lambda2))) * 0.5)), cos(phi1), ((phi1 * phi1) * 0.25))), 0.5)));
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(R * Float64(2.0 * atan(sqrt(fma(cos(phi1), (sin(Float64(0.5 * Float64(lambda1 - lambda2))) ^ 2.0), Float64(0.25 * (phi1 ^ 2.0)))), (Float64(1.0 - fma(Float64(0.5 - Float64(cos(Float64(1.0 * Float64(lambda1 - lambda2))) * 0.5)), cos(phi1), Float64(Float64(phi1 * phi1) * 0.25))) ^ 0.5))))
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[Cos[phi1], $MachinePrecision] * N[Power[N[Sin[N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[(0.25 * N[Power[phi1, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Power[N[(1.0 - N[(N[(0.5 - N[(N[Cos[N[(1.0 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision] + N[(N[(phi1 * phi1), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.0%

   \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

2. Taylor expanded in phi2 around 0

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
3. Step-by-step derivation

   1. lower-fma.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. lower-cos.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. lower-pow.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. lower-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. lower-*.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   6. lower--.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. lower-pow.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   8. lower-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. lower-*.f64 46.7

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

4. Applied rewrites 46.7%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

5. Taylor expanded in phi2 around 0

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
6. Step-by-step derivation

   1. lower-fma.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   2. lower-cos.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   3. lower-pow.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   4. lower-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   5. lower-*.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   6. lower--.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   7. lower-pow.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   8. lower-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   9. lower-*.f64 46.8

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

7. Applied rewrites 46.8%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]

8. Taylor expanded in phi1 around 0

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]
9. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   2. lower-pow.f64 31.8

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

10. Applied rewrites 31.8%

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

11. Taylor expanded in phi1 around 0

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}\right) \]
12. Step-by-step derivation

    1. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}\right) \]

    2. lower-pow.f64 22.3

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}\right) \]

13. Applied rewrites 22.3%

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}\right) \]

15. Applied rewrites 27.5%

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\color{blue}{{\left(1 - \mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot 0.25\right)\right)}^{0.5}}}\right) \]
16. Add Preprocessing

Alternative 42: 27.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := \left(\phi_1 \cdot \phi_1\right) \cdot 0.25\\ t_2 := \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, t\_1\right)\\ t_3 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_0\right) \cdot t\_0\\ \mathbf{if}\;2 \cdot \tan^{-1}_* \frac{\sqrt{t\_3}}{\sqrt{1 - t\_3}} \leq 0.1:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_2}}{\sqrt{1 - t\_2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, \cos \phi_1, t\_1\right)}}{{\left(1 - \mathsf{fma}\left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5, \cos \phi_1, t\_1\right)\right)}^{0.5}} \cdot 2\right) \cdot R\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_1 (* (* phi1 phi1) 0.25))
        (t_2 (fma (cos phi1) (pow (sin (* 0.5 (- lambda1 lambda2))) 2.0) t_1))
        (t_3
         (+
          (pow (sin (/ (- phi1 phi2) 2.0)) 2.0)
          (* (* (* (cos phi1) (cos phi2)) t_0) t_0))))
   (if (<= (* 2.0 (atan2 (sqrt t_3) (sqrt (- 1.0 t_3)))) 0.1)
     (* R (* 2.0 (atan2 (sqrt t_2) (sqrt (- 1.0 t_2)))))
     (*
      (*
       (atan2
        (sqrt
         (fma
          (- 0.5 (* (cos (* 1.0 (- lambda1 lambda2))) 0.5))
          (cos phi1)
          t_1))
        (pow
         (- 1.0 (fma (- 0.5 (* (cos (- lambda2 lambda1)) 0.5)) (cos phi1) t_1))
         0.5))
       2.0)
      R))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(((lambda1 - lambda2) / 2.0));
	double t_1 = (phi1 * phi1) * 0.25;
	double t_2 = fma(cos(phi1), pow(sin((0.5 * (lambda1 - lambda2))), 2.0), t_1);
	double t_3 = pow(sin(((phi1 - phi2) / 2.0)), 2.0) + (((cos(phi1) * cos(phi2)) * t_0) * t_0);
	double tmp;
	if ((2.0 * atan2(sqrt(t_3), sqrt((1.0 - t_3)))) <= 0.1) {
		tmp = R * (2.0 * atan2(sqrt(t_2), sqrt((1.0 - t_2))));
	} else {
		tmp = (atan2(sqrt(fma((0.5 - (cos((1.0 * (lambda1 - lambda2))) * 0.5)), cos(phi1), t_1)), pow((1.0 - fma((0.5 - (cos((lambda2 - lambda1)) * 0.5)), cos(phi1), t_1)), 0.5)) * 2.0) * R;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_1 = Float64(Float64(phi1 * phi1) * 0.25)
	t_2 = fma(cos(phi1), (sin(Float64(0.5 * Float64(lambda1 - lambda2))) ^ 2.0), t_1)
	t_3 = Float64((sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0) + Float64(Float64(Float64(cos(phi1) * cos(phi2)) * t_0) * t_0))
	tmp = 0.0
	if (Float64(2.0 * atan(sqrt(t_3), sqrt(Float64(1.0 - t_3)))) <= 0.1)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_2), sqrt(Float64(1.0 - t_2)))));
	else
		tmp = Float64(Float64(atan(sqrt(fma(Float64(0.5 - Float64(cos(Float64(1.0 * Float64(lambda1 - lambda2))) * 0.5)), cos(phi1), t_1)), (Float64(1.0 - fma(Float64(0.5 - Float64(cos(Float64(lambda2 - lambda1)) * 0.5)), cos(phi1), t_1)) ^ 0.5)) * 2.0) * R);
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(phi1 * phi1), $MachinePrecision] * 0.25), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[phi1], $MachinePrecision] * N[Power[N[Sin[N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[(N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(2.0 * N[ArcTan[N[Sqrt[t$95$3], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$3), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.1], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$2], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$2), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[ArcTan[N[Sqrt[N[(N[(0.5 - N[(N[Cos[N[(1.0 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision] / N[Power[N[(1.0 - N[(N[(0.5 - N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * R), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 2 binary64) (atan2.f64 (sqrt.f64 (+.f64 (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) #s(literal 2 binary64))) #s(literal 2 binary64)) (*.f64 (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))))) (sqrt.f64 (-.f64 #s(literal 1 binary64) (+.f64 (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) #s(literal 2 binary64))) #s(literal 2 binary64)) (*.f64 (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64))))))))) < 0.10000000000000001

   1. Initial program 62.0%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-*.f64 46.7

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 46.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   6. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      9. lower-*.f64 46.8

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 46.8%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]

   8. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      2. lower-pow.f64 31.8

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   10. Applied rewrites 31.8%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   11. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}\right) \]
   12. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}\right) \]

       2. lower-pow.f64 22.3

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}\right) \]

   13. Applied rewrites 22.3%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}\right) \]
   14. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}\right) \]

       2. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\phi_1}^{2} \cdot \frac{1}{4}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}\right) \]

       3. lift-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\phi_1}^{2} \cdot \frac{1}{4}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}\right) \]

       4. unpow2 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}\right) \]

       5. lower-*.f32 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}\right) \]

       6. lower-unsound-*.f32 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}\right) \]

       8. lower-unsound-*.f64 22.3

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \left(\phi_1 \cdot \phi_1\right) \cdot 0.25\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}\right) \]

   15. Applied rewrites 22.3%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \left(\phi_1 \cdot \phi_1\right) \cdot 0.25\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}\right) \]
   16. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}\right) \]

       2. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\phi_1}^{2} \cdot \frac{1}{4}\right)}}\right) \]

       3. lift-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\phi_1}^{2} \cdot \frac{1}{4}\right)}}\right) \]

       4. unpow2 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}}\right) \]

       5. lower-*.f32 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}}\right) \]

       6. lower-unsound-*.f32 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}}\right) \]

       8. lower-unsound-*.f64 22.3

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \left(\phi_1 \cdot \phi_1\right) \cdot 0.25\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \left(\phi_1 \cdot \phi_1\right) \cdot 0.25\right)}}\right) \]

   17. Applied rewrites 22.3%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \left(\phi_1 \cdot \phi_1\right) \cdot 0.25\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \left(\phi_1 \cdot \phi_1\right) \cdot 0.25\right)}}\right) \]

   if 0.10000000000000001 < (*.f64 #s(literal 2 binary64) (atan2.f64 (sqrt.f64 (+.f64 (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) #s(literal 2 binary64))) #s(literal 2 binary64)) (*.f64 (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))))) (sqrt.f64 (-.f64 #s(literal 1 binary64) (+.f64 (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) #s(literal 2 binary64))) #s(literal 2 binary64)) (*.f64 (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))))))))

   1. Initial program 62.0%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-*.f64 46.7

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 46.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   6. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      9. lower-*.f64 46.8

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 46.8%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]

   8. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      2. lower-pow.f64 31.8

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   10. Applied rewrites 31.8%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   11. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}\right) \]
   12. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}\right) \]

       2. lower-pow.f64 22.3

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}\right) \]

   13. Applied rewrites 22.3%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}\right) \]

   15. Applied rewrites 19.9%

       \[\leadsto \color{blue}{\left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot 0.25\right)}}{\sqrt{1 - \mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot 0.25\right)}} \cdot 2\right) \cdot R} \]

   17. Applied rewrites 25.0%

       \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot 0.25\right)}}{\color{blue}{{\left(1 - \mathsf{fma}\left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot 0.25\right)\right)}^{0.5}}} \cdot 2\right) \cdot R \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 43: 25.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} t_0 := \left(\phi_1 \cdot \phi_1\right) \cdot 0.25\\ \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, \cos \phi_1, t\_0\right)}}{{\left(1 - \mathsf{fma}\left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5, \cos \phi_1, t\_0\right)\right)}^{0.5}} \cdot 2\right) \cdot R \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (* phi1 phi1) 0.25)))
   (*
    (*
     (atan2
      (sqrt
       (fma (- 0.5 (* (cos (* 1.0 (- lambda1 lambda2))) 0.5)) (cos phi1) t_0))
      (pow
       (- 1.0 (fma (- 0.5 (* (cos (- lambda2 lambda1)) 0.5)) (cos phi1) t_0))
       0.5))
     2.0)
    R)))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = (phi1 * phi1) * 0.25;
	return (atan2(sqrt(fma((0.5 - (cos((1.0 * (lambda1 - lambda2))) * 0.5)), cos(phi1), t_0)), pow((1.0 - fma((0.5 - (cos((lambda2 - lambda1)) * 0.5)), cos(phi1), t_0)), 0.5)) * 2.0) * R;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(Float64(phi1 * phi1) * 0.25)
	return Float64(Float64(atan(sqrt(fma(Float64(0.5 - Float64(cos(Float64(1.0 * Float64(lambda1 - lambda2))) * 0.5)), cos(phi1), t_0)), (Float64(1.0 - fma(Float64(0.5 - Float64(cos(Float64(lambda2 - lambda1)) * 0.5)), cos(phi1), t_0)) ^ 0.5)) * 2.0) * R)
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[(phi1 * phi1), $MachinePrecision] * 0.25), $MachinePrecision]}, N[(N[(N[ArcTan[N[Sqrt[N[(N[(0.5 - N[(N[Cos[N[(1.0 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision] / N[Power[N[(1.0 - N[(N[(0.5 - N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * R), $MachinePrecision]]

Derivation

1. Initial program 62.0%

   \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

2. Taylor expanded in phi2 around 0

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
3. Step-by-step derivation

   1. lower-fma.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. lower-cos.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. lower-pow.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. lower-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. lower-*.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   6. lower--.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. lower-pow.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   8. lower-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. lower-*.f64 46.7

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

4. Applied rewrites 46.7%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

5. Taylor expanded in phi2 around 0

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
6. Step-by-step derivation

   1. lower-fma.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   2. lower-cos.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   3. lower-pow.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   4. lower-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   5. lower-*.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   6. lower--.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   7. lower-pow.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   8. lower-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   9. lower-*.f64 46.8

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

7. Applied rewrites 46.8%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]

8. Taylor expanded in phi1 around 0

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]
9. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   2. lower-pow.f64 31.8

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

10. Applied rewrites 31.8%

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

11. Taylor expanded in phi1 around 0

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}\right) \]
12. Step-by-step derivation

    1. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}\right) \]

    2. lower-pow.f64 22.3

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}\right) \]

13. Applied rewrites 22.3%

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}\right) \]

15. Applied rewrites 19.9%

    \[\leadsto \color{blue}{\left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot 0.25\right)}}{\sqrt{1 - \mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot 0.25\right)}} \cdot 2\right) \cdot R} \]

17. Applied rewrites 25.0%

    \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot 0.25\right)}}{\color{blue}{{\left(1 - \mathsf{fma}\left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot 0.25\right)\right)}^{0.5}}} \cdot 2\right) \cdot R \]
18. Add Preprocessing

Alternative 44: 19.9% accurate, 2.4× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, 1 + -0.5 \cdot {\phi_1}^{2}, \left(\phi_1 \cdot \phi_1\right) \cdot 0.25\right)\\ \left(\tan^{-1}_* \frac{\sqrt{t\_0}}{\sqrt{1 - t\_0}} \cdot 2\right) \cdot R \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (fma
          (- 0.5 (* (cos (* 1.0 (- lambda1 lambda2))) 0.5))
          (+ 1.0 (* -0.5 (pow phi1 2.0)))
          (* (* phi1 phi1) 0.25))))
   (* (* (atan2 (sqrt t_0) (sqrt (- 1.0 t_0))) 2.0) R)))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = fma((0.5 - (cos((1.0 * (lambda1 - lambda2))) * 0.5)), (1.0 + (-0.5 * pow(phi1, 2.0))), ((phi1 * phi1) * 0.25));
	return (atan2(sqrt(t_0), sqrt((1.0 - t_0))) * 2.0) * R;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = fma(Float64(0.5 - Float64(cos(Float64(1.0 * Float64(lambda1 - lambda2))) * 0.5)), Float64(1.0 + Float64(-0.5 * (phi1 ^ 2.0))), Float64(Float64(phi1 * phi1) * 0.25))
	return Float64(Float64(atan(sqrt(t_0), sqrt(Float64(1.0 - t_0))) * 2.0) * R)
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[(0.5 - N[(N[Cos[N[(1.0 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(-0.5 * N[Power[phi1, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(phi1 * phi1), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[ArcTan[N[Sqrt[t$95$0], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * R), $MachinePrecision]]

Derivation

1. Initial program 62.0%

   \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

2. Taylor expanded in phi2 around 0

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
3. Step-by-step derivation

   1. lower-fma.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. lower-cos.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. lower-pow.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. lower-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. lower-*.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   6. lower--.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. lower-pow.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   8. lower-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. lower-*.f64 46.7

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

4. Applied rewrites 46.7%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

5. Taylor expanded in phi2 around 0

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
6. Step-by-step derivation

   1. lower-fma.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   2. lower-cos.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   3. lower-pow.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   4. lower-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   5. lower-*.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   6. lower--.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   7. lower-pow.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   8. lower-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   9. lower-*.f64 46.8

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

7. Applied rewrites 46.8%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]

8. Taylor expanded in phi1 around 0

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]
9. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   2. lower-pow.f64 31.8

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

10. Applied rewrites 31.8%

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

11. Taylor expanded in phi1 around 0

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}\right) \]
12. Step-by-step derivation

    1. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}\right) \]

    2. lower-pow.f64 22.3

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}\right) \]

13. Applied rewrites 22.3%

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}\right) \]

15. Applied rewrites 19.9%

    \[\leadsto \color{blue}{\left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot 0.25\right)}}{\sqrt{1 - \mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot 0.25\right)}} \cdot 2\right) \cdot R} \]

16. Taylor expanded in phi1 around 0

    \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, 1 + \color{blue}{\frac{-1}{2} \cdot {\phi_1}^{2}}, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}} \cdot 2\right) \cdot R \]
17. Step-by-step derivation

    1. lower-+.f64 N/A

       \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, 1 + \frac{-1}{2} \cdot \color{blue}{{\phi_1}^{2}}, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}} \cdot 2\right) \cdot R \]

    2. lower-*.f64 N/A

       \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, 1 + \frac{-1}{2} \cdot {\phi_1}^{\color{blue}{2}}, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}} \cdot 2\right) \cdot R \]

    3. lower-pow.f64 19.9

       \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, 1 + -0.5 \cdot {\phi_1}^{2}, \left(\phi_1 \cdot \phi_1\right) \cdot 0.25\right)}}{\sqrt{1 - \mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot 0.25\right)}} \cdot 2\right) \cdot R \]

18. Applied rewrites 19.9%

    \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, 1 + \color{blue}{-0.5 \cdot {\phi_1}^{2}}, \left(\phi_1 \cdot \phi_1\right) \cdot 0.25\right)}}{\sqrt{1 - \mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot 0.25\right)}} \cdot 2\right) \cdot R \]

19. Taylor expanded in phi1 around 0

    \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, 1 + \frac{-1}{2} \cdot {\phi_1}^{2}, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, 1 + \color{blue}{\frac{-1}{2} \cdot {\phi_1}^{2}}, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}} \cdot 2\right) \cdot R \]
20. Step-by-step derivation

    1. lower-+.f64 N/A

       \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, 1 + \frac{-1}{2} \cdot {\phi_1}^{2}, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, 1 + \frac{-1}{2} \cdot \color{blue}{{\phi_1}^{2}}, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}} \cdot 2\right) \cdot R \]

    2. lower-*.f64 N/A

       \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, 1 + \frac{-1}{2} \cdot {\phi_1}^{2}, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, 1 + \frac{-1}{2} \cdot {\phi_1}^{\color{blue}{2}}, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}} \cdot 2\right) \cdot R \]

    3. lower-pow.f64 19.9

       \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, 1 + -0.5 \cdot {\phi_1}^{2}, \left(\phi_1 \cdot \phi_1\right) \cdot 0.25\right)}}{\sqrt{1 - \mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, 1 + -0.5 \cdot {\phi_1}^{2}, \left(\phi_1 \cdot \phi_1\right) \cdot 0.25\right)}} \cdot 2\right) \cdot R \]

21. Applied rewrites 19.9%

    \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, 1 + -0.5 \cdot {\phi_1}^{2}, \left(\phi_1 \cdot \phi_1\right) \cdot 0.25\right)}}{\sqrt{1 - \mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, 1 + \color{blue}{-0.5 \cdot {\phi_1}^{2}}, \left(\phi_1 \cdot \phi_1\right) \cdot 0.25\right)}} \cdot 2\right) \cdot R \]
22. Add Preprocessing

Reproduce

herbie shell --seed 2025178 
(FPCore (R lambda1 lambda2 phi1 phi2)
  :name "Distance on a great circle"
  :precision binary64
  (* R (* 2.0 (atan2 (sqrt (+ (pow (sin (/ (- phi1 phi2) 2.0)) 2.0) (* (* (* (cos phi1) (cos phi2)) (sin (/ (- lambda1 lambda2) 2.0))) (sin (/ (- lambda1 lambda2) 2.0))))) (sqrt (- 1.0 (+ (pow (sin (/ (- phi1 phi2) 2.0)) 2.0) (* (* (* (cos phi1) (cos phi2)) (sin (/ (- lambda1 lambda2) 2.0))) (sin (/ (- lambda1 lambda2) 2.0))))))))))